Monodromy and vanishing cycles in toric surfaces

Abstract

Given an ample line bundle on a toric surface, a question of Donaldson asks which simple closed curves can be vanishing cycles for nodal degenerations of smooth curves in the complete linear system. This paper provides a complete answer. This is accomplished by reformulating the problem in terms of the mapping class group-valued monodromy of the linear system, and giving a precise determination of this monodromy group.

1 Introduction

Let X be a smooth toric surface and ${\mathcal {L}}$ an ample line bundle on X. In the complete linear system $\left| {\mathcal {L}} \right|$, there is a hypersurface ${\mathcal {D}}$ known as the discriminant locus consisting of the singular curves $C \in \left| {\mathcal {L}} \right|$. The complement

$$\begin{aligned} {\mathcal {M}}({\mathcal {L}}) \,{:}{=}\, \left| {\mathcal {L}} \right| {\setminus } {\mathcal {D}} \end{aligned}$$

therefore supports a tautological family of closed Riemann surfaces of some genus $g({\mathcal {L}})$. Topologically, this is a fiber bundle $\pi {:}\,{\mathcal {E}}({\mathcal {L}}) \rightarrow {\mathcal {M}}({\mathcal {L}})$ with fiber $\Sigma _{g({\mathcal {L}})}$. Consequently, there is a monodromy representation

$$\begin{aligned} \mu _{{\mathcal {L}}}{:}\,\pi _1({\mathcal {M}}({\mathcal {L}}),C_0) \rightarrow {{\,\mathrm{Mod}\,}}(C_0). \end{aligned}$$

Here, $C_0 \in {\mathcal {M}}({\mathcal {L}})$ is a fixed curve, and ${{\,\mathrm{Mod}\,}}(C_0)\,{:}{=}\, \pi _0({{\,\mathrm{Diff}\,}}^+(C_0))$ denotes the mapping class group of $C_0$ (see Sect. 2.1). Under $\mu _{\mathcal {L}}$, a based loop $\gamma \in \pi _1({\mathcal {M}}(\mathcal L),C_0)$ is mapped to (the isotopy class of) the diffeomorphism $\mu _{\mathcal {L}}(\gamma ) \in {{\,\mathrm{Diff}\,}}(C_0)$ obtained by “parallel transport” of $C_0$ along $\gamma$. For details, see, e.g., [7, Section 5.6.1].

In this paper, we give a nearly complete answer to the following fundamental question. Define

$$\begin{aligned} \Gamma _{{\mathcal {L}}} \,{:}{=}\, {{\,\mathrm{Im}\,}}(\mu _{{\mathcal {L}}}) \leqslant {{\,\mathrm{Mod}\,}}\left( \Sigma _{g({\mathcal {L}})}\right) . \end{aligned}$$

Question 1.1

What is $\Gamma _{{\mathcal {L}}}$? When is it a finite-index subgroup of ${{\,\mathrm{Mod}\,}}(\Sigma _{g({\mathcal {L}})})$? Can one give a precise characterization of $\Gamma _{{\mathcal {L}}}$?

Question 1.1 is closely related to a question posed by Donaldson [6]. Fix a curve $C_0 \in \mathcal M({\mathcal {L}})$ and an identification $C_0 \cong \Sigma _{g(\mathcal L)}$. Define a vanishing cycle for ${\mathcal {L}}$ as a simple closed curve $\gamma$ on $C_0$ for which there is a degeneration of $C_0$ to a curve $C'$ with a single node, such that $\gamma$ becomes null-homotopic on $C'$. If c is a vanishing cycle, then necessarily the Dehn twist $T_c$ lies in $\Gamma _{{\mathcal {L}}}$; it arises from a loop in ${\mathcal {M}}({\mathcal {L}})$ encircling the nodal curve in $\left| {\mathcal {L}} \right|$.

Question 1.2

(Donaldson) For ${\mathcal {L}}$ an ample line bundle on a smooth toric surface X, which curves (on a fixed $C_0)$ are vanishing cycles?

A first insight into Questions 1.1 and 1.2 is to observe the presence of an invariant “higher spin structure”. Let $K_X$ denote the canonical bundle of X. The adjoint line bundle of ${\mathcal {L}}$ is the line bundle ${\mathcal {L}} \otimes K_X$. Define $r \in {\mathbb {N}}$ to be the highest root of ${\mathcal {L}} \otimes K_X$ in ${{\,\mathrm{Pic}\,}}(X)$. As explained in Proposition 10.2, associated to ${\mathcal {L}} \otimes K_X$ is a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure$\phi _{{\mathcal {L}}}$, and the associated stabilizer subgroup ${{\,\mathrm{Mod}\,}}(\Sigma _{g({\mathcal {L}})})[\phi _{{\mathcal {L}}}]$ (see Definition 3.14). Proposition 10.2 asserts that necessarily $\Gamma _{\mathcal {L}} \leqslant {{\,\mathrm{Mod}\,}}(\Sigma _{g({\mathcal {L}})})[\phi _{{\mathcal {L}}}]$. The function $\phi _{\mathcal {L}}$ gives rise to a notion of admissible curve and the associated subgroup ${\mathcal {T}}_{\phi _{{\mathcal {L}}}} \leqslant {{\,\mathrm{Mod}\,}}(\Sigma _{g({\mathcal {L}})})[\phi _{{\mathcal {L}}}]$ of admissible twists (see Definition 3.16). If a curve c is a vanishing cycle, it is necessarily admissible; see Lemma 3.15. Our main theorem asserts that these necessary conditions are also sufficient (at least “virtually” so, in the case r is even).

Theorem A

Let ${\mathcal {L}}$ be an ample line bundle on a smooth toric surface X for which the generic fiber is not hyperelliptic. Assume $r > 1$ or else $g({\mathcal {L}}) \ge 5$.

• If r is odd, then $\Gamma _{{\mathcal {L}}} = {{\,\mathrm{Mod}\,}}(\Sigma _{g({\mathcal {L}})})[\phi _{{\mathcal {L}}}]$.

• If r is even, then $\Gamma _{{\mathcal {L}}} \leqslant {{\,\mathrm{Mod}\,}}(\Sigma _{g({\mathcal {L}})})$ is a finite-index subgroup that contains ${\mathcal {T}}_{\phi _{{\mathcal {L}}}}$.

In either case, $[{{\,\mathrm{Mod}\,}}(\Sigma _{g({\mathcal {L}})}):\Gamma _{{\mathcal {L}}}]$ is finite. Moreover, Question 1.2 admits the following complete answer: a curve$\gamma$is a vanishing cycle if and only if$\gamma$is an admissible curve.

We remark that many familiar algebraic surfaces such as ${{\mathbb {CP}}}^2$ and ${{\mathbb {CP}}}^1 \times {{\mathbb {CP}}}^1$ are smooth toric surfaces. For instance, as a special case of Theorem A we obtain the following theorem concerning smooth plane curves. The case $d = 5$ was addressed in [17], while the cases $d \le 4$ are either classical or trivial.

Theorem 1.3

Set $g = {d-1 \atopwithdelims ()2}$, and define

$$\begin{aligned} \Gamma _d \leqslant {{\,\mathrm{Mod}\,}}(\Sigma _g) \end{aligned}$$

to be the monodromy group of the family of smooth curves in ${{\mathbb {CP}}}^2$ of degree d, i.e. the group $\Gamma _{\mathcal {L}}$ for the line bundle ${\mathcal {L}} = {\mathcal {O}}(d)$ on ${{\mathbb {CP}}}^2$. Then there exists a ${\mathbb {Z}}/(d-3){\mathbb {Z}}$-valued spin structure $\phi _d$ such that the following hold.

• If d is even, then $\Gamma _d = {{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi _d]$.

• If d is odd, then $\Gamma _d$ is of finite index in ${{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi _d]$, where $\Gamma _d$ contains the subgroup ${\mathcal {T}}_{\phi _d}$ of admissible twists.

Theorem A also addresses a conjecture that was independently formulated by the author in [17] in the case of $X = {{\mathbb {CP}}}^2$, and in full generality by Crétois and Lang [3].

Conjecture 1.4

For any pair $(X, {\mathcal {L}})$ as above, there is an equality

$$\begin{aligned} \Gamma _{{\mathcal {L}}} = {{\,\mathrm{Mod}\,}}(\Sigma _{g({\mathcal {L}})})[\phi _{\mathcal L}]. \end{aligned}$$

Theorem A resolves Conjecture 1.4 in the affirmative whenever r is odd, and shows that in the case r even, $\Gamma _{{\mathcal {L}}}$ is at least of finite index in ${{\,\mathrm{Mod}\,}}(\Sigma _{g({\mathcal {L}})})[\phi _{\mathcal L}]$.

Theorem A is proved using a combination of methods from toric geometry and the theory of the mapping class group. On the toric end of the spectrum, we make essential use of the powerful results developed by Crétois and Lang [3]. The centerpiece of their theory is a combinatorial model for a curve $C_0 \in {\mathcal {M}}({\mathcal {L}})$ based around a convex lattice polygon. Their results give a description of vanishing cycles in terms of lattice points and line segments, and allow one to produce many elements of $\Gamma _{{\mathcal {L}}}$. Crétois–Lang developed their methods in order to address Question 1.2 and Conjecture 1.4 in the case $r \le 2$, and obtained complete answers in these cases. See [3] for the case $r = 1$, and [4] for the case $r = 2$, as well as the case where the general fiber is hyperelliptic.

On the mapping class group side, we carry out an extensive investigation of the groups ${{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ]$ and $\mathcal T_\phi$ mentioned above. We remark here that the theory of higher spin structures does not require the presence of a specific ample line bundle ${\mathcal {L}}$, and so we adjust notation accordingly and refer to Riemann surfaces $\Sigma _g$, spin structures $\phi$, etc. Our main result here is a general criterion for a collection of Dehn twists to generate (a finite-index subgroup of) ${{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ]$, given in Theorem 9.5.

Outline of the paper The bulk of the paper (Sects. 2–9) is devoted to developing the mapping class group technology necessary to show that the vanishing cycles investigated by Crétois–Lang generate a finite-index subgroup of the mapping class group. This culminates in Theorem 9.5. Portions of Theorem 9.5 are established earlier in Propositions 5.1 and 6.2.

Sections 2–4 contain preliminary results that are used throughout the paper. Section 2 collects the necessary background on mapping class groups; these results are standard and are included so as to fix notation and terminology, and to serve as a guide to the reader approaching the paper from a background in toric geometry. Section 3 presents the basic theory of higher spin structures, building off the foundational work of Humphries and Johnson [9]. Section 4 describes the action of the mapping class group on the set of higher spin structures. This yields several crucial corollaries (Corollaries 4.5, 4.10, 4.11) concerning the existence of configurations of curves with prescribed properties which are used extensively in subsequent sections.

Theorem 9.5 gives a criterion for a collection of Dehn twists to generate the so-called admissible subgroup${\mathcal {T}}_\phi$ associated to a higher spin structure $\phi$. A study of the admissible subgroup is sufficient to answer Question 1.2. The reader interested only in this portion of Theorem A can skip Sects. 5 and 6 and jump directly from Sects. 4–7.

The proof of Theorem 9.5 is carried out in Sects. 7–9. Section 7 establishes the connectivity of certain simplicial complexes acted on by the stabilizer subgroup of a higher spin structure. These results are used in the argument of Sect. 8, and also underlie the method by which the admissible subgroup is used to study the set of vanishing cycles. Section 8 is devoted to a study of certain subgroups of the admissible subgroup; the main result Proposition 8.2 furnishes a generating set for $\mathcal T_\phi$ in terms of these subgroups. Section 9 introduces the notion of a network; ultimately a network is a technical device used to factor the generators given in Proposition 8.2 into products of Dehn twists. Theorem 9.5 gives a sufficient condition, formulated in the language of networks, for a collection of Dehn twists to generate a subgroup containing the admissible subgroup.

The portion of Theorem A that goes beyond Question 1.2 concerns establishing that the admissible subgroup is finite-index in the mapping class group. This is the content of Sects. 5 and 6, which treat the case where the ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure under study has r odd or even, respectively. The arguments for these two cases are substantially different, owing to the fact that in the case of r even, the higher spin structure has an Arf invariant which must be accounted for in various guises.

The net result of Sects. 2–9 is a criterion for a finite collection of Dehn twists to generate a finite-index subgroup of the mapping class group. In the final two sections, these results are applied in the setting of monodromy groups of linear systems on toric surfaces. Section 10 contains the necessary background material on toric surfaces, concentrating on the work of Crétois–Lang describing a particular finite collection of vanishing cycles. Section 11 exhibits a network amongst the set of vanishing cycles discussed in Sect. 10 and verifies that this network satisfies the hypotheses of Theorem 9.5 in order to obtain Theorem A.

2 Mapping class groups

This section collects background material on mapping class groups that will be used throughout the arguments in Sects. 3–9. Most of the material can be found in [7] and so will only be touched on briefly. The exception to this is the $D_n$ relation of Sect. 2.3, which will consequently be dealt with in greater detail.

2.1 Basics

The material in this section is almost certainly well-known to a reader conversant in mapping class groups, but is included so as to fix notation and terminology.

Genus, boundary, punctures All surfaces under consideration are oriented and of finite type. A surface of genus g with n punctures and b boundary components is denoted by $\Sigma _{g,b}^n$. When one or more of $b, n = 0$, the corresponding decoration will be omitted.

Intersection numbers Let a, b be simple closed curves on a surface S. Often we will confuse the distinction between a simple closed curve and its isotopy class. The geometric intersection number between a, b will be notated i(a, b) (see [7, Section 1.2.3]). For oriented simple closed curves a, b, the algebraic intersection number is denoted $\langle a,b \rangle$. Of course, algebraic intersection depends only on the homology classes $[a],[b] \in H_1(S; {\mathbb {Z}})$.

Mapping class groups Let $\Sigma _{g,b}^n$ be a surface. The mapping class group of $\Sigma _{g,b}^n$, written ${{\,\mathrm{Mod}\,}}(\Sigma _{g,b}^n)$, is defined as

$$\begin{aligned} {{\,\mathrm{Mod}\,}}(\Sigma _{g,b}^n) \,{:}{=}\, \pi _0\left( {{\,\mathrm{Diff}\,}}^+(\Sigma _{g,b}^n, \partial \Sigma _{g,b}^n)\right) , \end{aligned}$$

where ${{\,\mathrm{Diff}\,}}^+(\Sigma _{g,b}^n, \partial \Sigma _{g,b}^n)$ denotes the group of orientation-preserving diffeomorphisms of $\Sigma _{g,b}^n$ that restrict to the identity on the boundary of $\Sigma _{g,b}^n$ and fix the punctures pointwise (not merely setwise, as some authors adopt).

The standard generators For a simple closed curve a on $\Sigma _{g,b}^n$, the left-handed Dehn twist about a is written $T_a$. For $g \ge 2$, the standard generators form a generating set for ${{\,\mathrm{Mod}\,}}(\Sigma _{g,b}^n)$ consisting of the Dehn twists about the curves $a_1, a_2, b_1, \dots , b_g, c_1, \dots , c_{g-1}, d_1, \dots , d_{b+n-1}$ shown in Fig. 1.

Fig. 1

The standard generators for $\Sigma _{g,3}^3$

The change-of-coordinates principle The classification of surfaces theorem asserts that if $S, S'$ are two (connected and orientable) surfaces of finite type with the same genus, number of punctures, and number of boundary components, then there is a diffeomorphism $f{:}\,S \rightarrow S'$. This is often exploited in the study of mapping class groups in the guise of the “change-of-coordinates principle”. It is difficult to write down a single, all-encompassing statement of the change-of-coordinates principle, but informally, it states that any configuration of curves, arcs, and/or subsurfaces of a surfaceSis determined up to diffeomorphism by combinatorial information alone. In the present paper, the change-of-coordinates principle will often be invoked tacitly. The reader interested in a more thorough discussion of the change-of-coordinates principle is referred to [7, Section 1.3].

One consequence of the change-of-coordinates principle is that it becomes easy to understand the ${{\,\mathrm{Mod}\,}}(S)$ orbits of many different kinds of configurations. As an example, we discuss here the action on geometric symplectic bases for S.

Definition 2.1

Let S be a surface of genus $g \ge 0$ with $n\ge 0$ boundary components and $b \ge 0$ punctures. A geometric symplectic basis for S is a collection of oriented simple closed curves ${\mathcal {B}} = \{\alpha _1, \beta _1, \dots , \alpha _g, \beta _g\}$ satisfying the following properties:

• (1) $i(a_i, b_i) = 1$ for each $i = 1, \dots , g$, and all other pairs of elements of ${\mathcal {B}}$ are disjoint,

• (2) $\langle [a_i], [b_i] \rangle = 1$ for each $i = 1, \dots , g$.

Remark 2.2

The (homology classes of the) curves in a geometric symplectic basis form a basis for $H_1(S;{\mathbb {Z}})$ in the sense of linear algebra only when $n + b \le 1$. In this paper, geometric symplectic bases are used to study ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structures. Proposition 3.8 and Theorem 3.9 together imply that a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure is determined by its “signature” (Definition 4.1) in combination with its values on a geometric symplectic basis.

The following is a typical statement that is proved using the change-of-coordinates principle.

Lemma 2.3

Let ${\mathcal {B}}$ and ${\mathcal {B}}'$ be two geometric symplectic bases for S. Then there is a diffeomorphism $f{:}\,S \rightarrow S$ such that $f({\mathcal {B}}) = {\mathcal {B}}'$.

2.2 The Birman exact sequence

A reference for this subsection is [7, Section 4.2]. Consider a surface $\Sigma _{g,b}^n$ with $n \ge 1$ and $2g+b+n \ge 4$. There is an inclusion $\Sigma _{g,b}^n \hookrightarrow \Sigma _{g,b}^{n-1}$ obtained by filling p in. This induces the Birman exact sequence

$$\begin{aligned} 1 \rightarrow \pi _1(\Sigma _{g,b}^{n-1}, p) \rightarrow {{\,\mathrm{Mod}\,}}(\Sigma _{g,b}^n) \rightarrow {{\,\mathrm{Mod}\,}}(\Sigma _{g,b}^{n-1}) \rightarrow 1. \end{aligned}$$

(1)

There is a slight variation on the Birman exact sequence where one fills in a boundary component with a closed disk, originally due to Johnson. In order to formulate this, we recall that the unit tangent bundle to a surface S is written UTS. Then the inclusion $\Sigma _{g,b}^n \rightarrow \Sigma _{g,b-1}^n$ induces the short exact sequence

$$\begin{aligned} 1 \rightarrow \pi _1(\textit{UT}\Sigma _{g,b-1}^n, {\tilde{p}}) \rightarrow {{\,\mathrm{Mod}\,}}(\Sigma _{g,b}^n) \rightarrow {{\,\mathrm{Mod}\,}}(\Sigma _{g,b-1}^n) \rightarrow 1, \end{aligned}$$

(2)

where ${\tilde{p}}$ is a unit tangent vector based at p. In both situations, the kernels admit descriptions in terms of Dehn twists. Consider first the case of (1). Let $\alpha$ be an embedded, oriented simple closed curve based at p, corresponding to an element $\alpha \in \pi _1(\Sigma _{g,b}^{n-1}, p)$. Let $\alpha _L$ (resp. $\alpha _R$) denote the left (resp. right) side of a neighborhood of $\alpha$. Both $\alpha _L, \alpha _R$ are simple closed curves on $\Sigma _{g,b}^n$. Then $\alpha \in \pi _1(\Sigma _{g,b}^{n-1}, p)$ corresponds to $T_{\alpha _L} T_{\alpha _R}^{-1} \in {{\,\mathrm{Mod}\,}}(\Sigma _{g,b}^n)$. The embedding $P{:}\,\pi _1(\Sigma _{g,b}^{n-1}, p) \rightarrow {{\,\mathrm{Mod}\,}}(\Sigma _{g,b}^n)$ is known as the point-pushing map, and $\pi _1(\Sigma _{g,b}^{n-1})$ is often referred to as the point-pushing subgroup of ${{\,\mathrm{Mod}\,}}(\Sigma _{g,b}^n)$.

It is a basic topological fact that for any surface $\Sigma _{g,b}^{n-1}$, there exists a collection of simple closed curves $\alpha _1, \dots , \alpha _k$ based at p, such that $\{\alpha _1, \dots , \alpha _k\}$ generates $\pi _1(\Sigma _{g,b}^{n-1},p)$. In practice, this means that to exhibit $\pi _1(\Sigma _{g,b}^n, p)$ as a subgroup of some group $H \leqslant {{\,\mathrm{Mod}\,}}(\Sigma _{g,b}^n)$, it suffices to exhibit this finite collection of multitwists.

In the case of (2), everything is much the same. Let $\Sigma _{g,b}^n \hookrightarrow \Sigma _{g,b-1}^n$ be an inclusion corresponding to capping off a boundary component $\Delta$ of $\Sigma _{g,b}^n$. Let $p \in \Sigma _{g,b-1}^n$ be a point on the interior of this new disk, and ${\tilde{p}}$ a tangent vector at p. Suppose that ${\tilde{\alpha }} \in \pi _1(\textit{UT}\Sigma _{g,b-1}^n,{\tilde{p}})$ corresponds to a framed simple closed curve $\alpha$ based at ${\tilde{p}}$. We define $\alpha _L$ and $\alpha _R$ as before. Then

$$\begin{aligned} P({\tilde{\alpha }}) = T_{\alpha _L} T_{\alpha _R}^{-1} T_\Delta ^k \end{aligned}$$

where $k \in {\mathbb {Z}}$ is the winding number of the tangent vector field specified by ${\tilde{\alpha }}$, relative to the tangential framing of the underlying curve $\alpha$. The subgroup $\pi _1(\textit{UT}\Sigma _{g,b-1}^n, {\tilde{p}})$ is known as the disk-pushing subgroup of ${{\,\mathrm{Mod}\,}}(\Sigma _{g,b}^n)$.

There is an analogous set of “geometric” generators for $\pi _1(\textit{UT}\Sigma _{g,b-1}^n, {\tilde{p}})$. Let $\alpha _1, \dots , \alpha _k$ be a collection of $C^1$-embedded simple closed curves on $\Sigma _{g,b-1}^n$ based at p such that $\pi _1(\Sigma _{g,b-1}^n,p) = \langle \alpha _1, \dots , \alpha _k \rangle$ as above. Each $\alpha _i$ determines an element ${\tilde{\alpha }}_i \in \pi _1(\Sigma _{g,b-1}^n, {\tilde{p}})$ via the so-called Johnson lift, whereby $\alpha _i$ is framed using the forward-pointing tangent vector. Suppose that each ${\tilde{\alpha }}_i$ is based at some common tangent vector ${\tilde{p}}$. Then $\pi _1(\textit{UT}\Sigma _{g,b-1}^n, {\tilde{p}})$ has a generating set of the following form:

$$\begin{aligned} \pi _1(\textit{UT}\Sigma _{g,b-1}^n, {\tilde{p}}) = \langle {\tilde{\alpha }}_1, \dots , {\tilde{\alpha }}_k, \zeta \rangle , \end{aligned}$$

where $\zeta$ is the loop around the $S^1$ fiber in the fibration $S^1 \rightarrow \textit{UT}\Sigma _{g,b-1}^n \rightarrow \Sigma _{g,b-1}^n$. In terms of Dehn twists, the Johnson lifts ${\tilde{\alpha }}_i$ correspond to mapping classes $T_{\alpha _{i,L}} T_{\alpha _{i,R}}^{-1}$ as before, while $\zeta$ corresponds to $T_\Delta$.

2.3 Relations

In this subsection we collect various relations in the mapping class group that will be used throughout the paper.

The braid relation Suppose a, b are simple closed curves satisfying $i(a,b) = 1$. Then the corresponding Dehn twists satisfy the braid relation:

$$\begin{aligned} T_a T_b T_a = T_b T_a T_b. \end{aligned}$$

We will also employ the following alternative form, formulated in terms of the curves a, b themselves:

$$\begin{aligned} T_a T_b (a) = b. \end{aligned}$$

The chain relation A chain of simple closed curves is a sequence $(a_1, \dots , a_n)$ of simple closed curves such that $i(a_i, a_{i+1}) = 1$ and $i(a_i, a_j) = 0$ otherwise. Let $\nu$ denote a regular neighborhood of a chain of length n, where the representative curves $a_1, \dots , a_n$ are in minimal position. When n is odd, $\partial \nu$ has two components $\Delta _1$ and $\Delta _2$; for n even, $\partial \nu = \Delta$ is a single (necessarily separating) curve. Abusing terminology, we will speak of the boundary of a chain itself, by which we mean the boundary of $\nu$. Given a subsurface S with 1 or 2 boundary components, a chain $a_1, \dots , a_n$ of curves on S is maximal if there is a deformation retraction of S onto $a_1 \cup \dots \cup a_n$. The following appears as [7, Proposition 4.12].

Proposition 2.4

(Chain relation) For n odd,

$$\begin{aligned} (T_{a_1} \dots T_{a_n})^{n+1} = T_{\Delta _1} T_{\Delta _2}, \end{aligned}$$

and for n even,

$$\begin{aligned} (T_{a_1} \dots T_{a_n})^{2n+2} = T_\Delta . \end{aligned}$$

Remark 2.5

The intersection pattern of a chain of n simple closed curves is recorded by the Dynkin diagram of type $A_n$, where vertices in the graph are adjacent if the corresponding curves intersect, and are nonadjacent if the curves are disjoint. Such a chain of curves determines a homomorphism from the Artin group $A(A_n)$ of type $A_n$ into the mapping class group ${{\,\mathrm{Mod}\,}}(\nu )$, where generators of $A(A_n)$ are sent to Dehn twists about the corresponding curves.

Under this homomorphism, the chain relation is a consequence of the fact that $A(A_n)$ has nontrivial center. The twist(s) about the boundary component(s) appearing on the right-hand side of the expressions in Proposition 2.4 are elements of the center of ${{\,\mathrm{Mod}\,}}(\nu )$, while the left-hand side merely gives the expression for a generator of $Z(A(A_n))$ as a word in the standard generators of $A(A_n)$. In [15, Section 2.4], Matsumoto explains how to determine the precise expression for this central element as a Dehn multitwist; this is the principle underlying the “$D_n$ relation” given in Proposition 2.6 below.

The${D_n}$relation There is an analogous (though less ubiquitous) relation that arises from a configuration of curves whose intersection pattern is modeled on the Dynkin diagram of type $D_n$. Proposition 2.6 below is the specialization of [15, Proposition 2.4] to the case of an Artin group of type $D_n$. The case of n odd is treated explicitly in [15, Theorem 1.5], while the case of n even is given an alternate proof in [17, Proposition 4.5].

Proposition 2.6

$(D_n$ relation) Let $n \ge 3$ be given, and express $n = 2g+1$ or $n = 2g+2$ according to whether n is odd or even. With reference to Fig. 2, let $H_n$ be the group generated by elements of the form $T_x$, with $x \in {\mathscr {D}}_n$ one of the curves below:

$$\begin{aligned} {\mathscr {D}}_{n} = \left\{ a, a', c_1, \dots , c_{n-2}\right\} . \end{aligned}$$

Then for $n = 2g+1$ odd,

$$\begin{aligned} T_{\Delta _0}^{2g-1} T_{\Delta _2} \in H_n, \end{aligned}$$

and for $n = 2g+2$ even,

$$\begin{aligned} T_{\Delta _0}^{g} T_{\Delta _1} T_{\Delta _1'}\in H_n. \end{aligned}$$

Fig. 2

The configuration of curves used in the $D_n$ relation. Note the presence of the unlabeled curve $c_{2g+1}$ on the far right side

The $D_n$ relation has some useful consequences which we record in Corollary 2.7 below. It is necessary to first describe the curves $C_k$ that will appear in the statement. For $1 \le k \le g+1$, let $\nu _k$ be a regular neighborhood of the subconfiguration ${\mathscr {D}}_{2k+1} \subset {\mathscr {D}}_n$. Each such $\nu _k$ is a surface of genus k with two boundary components. One of these is $\Delta _0$; the other is defined to be the curve $C_k$. Note in particular that $C_g = \Delta _2$ and that $C_{g+1}$ is the unlabeled boundary component of the ambient surface on the far right side of Fig. 2.

Corollary 2.7

Fix notation as in Proposition 2.6, and for $1 \le \ell \le 2g+3$, consider the configurations

$$\begin{aligned} {\mathscr {D}}_{\ell } = \left\{ a, a', c_1, \dots , c_{\ell -2}\right\} \end{aligned}$$

as in Fig. 2. Let $H_{2g+3}^+$ be the group generated by $H_{2g+3}$ and the Dehn twist $T_{\Delta _1}$. Then the following assertions hold:

• (1) $T_{\Delta _1'} \in H_{2g+3}^+$,

• (2) $T_{C_k}^m \in H_{2g+3}^+$ for any $1 \le k \le g+1$ and any m such that $(2k - 1)m$ divides g.

Proof

The proof of (1) follows from an important simple principle. Given a mapping class f and a simple closed curve d, there is a relation

$$\begin{aligned} f T_d f^{-1} = T_{f(d)}. \end{aligned}$$

It follows that if $f, T_d \in H_{2g+3}^+$, then also $T_{f(d)} \in H_{2g+3}^+$. To establish (1), we will find $f \in H_{2g+3}^+$ such that $f(c_{2g+1}) = \Delta _1'$. This will be accomplished by means of the braid relation.

The curves $a, a', c_1, \dots , c_{2g}$ are arranged in the configuration of the $D_{2g+2}$ relation; the boundary components correspond to $\Delta _0, \Delta _1, \Delta _1'$. By the $D_{2g+2}$-relation (Proposition 2.6),

$$\begin{aligned} T_{\Delta _0}^{g}T_{\Delta _1} T_{\Delta _1'} \in H_{2g+3}^+, \end{aligned}$$

and since $T_{\Delta _1} \in H_{2g+3}^+$ by assumption, also $T_{\Delta _0}^{g} T_{\Delta _1'} \in H_{2g+3}^+$. Since $\Delta _0$ is disjoint from both $c_{2g+1}$ and $\Delta _1'$, the braid relation implies that

$$\begin{aligned} T_{c_{2g+1}} T_{\Delta _0}^{g} T_{\Delta _1'}(c_{2g+1}) = T_{c_{2g+1}} T_{\Delta _1'} (c_{2g+1}) = \Delta _1'. \end{aligned}$$

Since $(T_{\Delta _0}^{g} T_{\Delta _1'})\in H_{2g+3}^+$, this shows $T_{\Delta _1'} \in H_{2g+3}^+$ as required.

We observe that (2) follows from the $D_{2k-1}$ relation (as applied to the subconfiguration ${\mathscr {D}}_{2k-1}$) and the claim that $T_{\Delta _0}^{g} \in H_{2g+3}^+$; this latter assertion follows from the $D_{2g+2}$ relation (applied to ${\mathscr {D}}_{2g+2}$) and (1). $\square$

2.4 The Torelli group

Most of the material in this subsection can be found in [7, Chapter 6], but see also [12]. We begin by observing that the action of ${{\,\mathrm{Mod}\,}}(\Sigma _g)$ on $H_1(\Sigma _g; {\mathbb {Z}})$ preserves the algebraic intersection pairing $\langle \cdot , \cdot \rangle$, leading to the symplectic representation

$$\begin{aligned} \Psi {:}\,{{\,\mathrm{Mod}\,}}(\Sigma _g) \rightarrow {{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}}). \end{aligned}$$

(3)

This is classically known to be a surjection. The Torelli group, notated ${\mathcal {I}}_g$, is the kernel of this representation:

$$\begin{aligned} {\mathcal {I}}_g \,{:}{=}\, \ker (\Psi ). \end{aligned}$$

Bounding pairs and separating twists There are two types of elements in ${\mathcal {I}}_g$ that will be of particular importance. Suppose that c, d are simple closed curves such that $c \cup d$ bounds a subsurface $S \cong \Sigma _{h,2}$. Then $T_c T_d^{-1} \in {\mathcal {I}}_g$ is known as a bounding pair map. The genus of a bounding pair map is slightly ambiguous: if $c \cup d$ bounds a surface $\Sigma _{h,2}$, then also $c \cup d$ bounds a surface $\Sigma _{g-h-1,2}$ on the other side. One defines the genus of $T_c T_d^{-1}$ as $\min \{h, g-h-1\}$. The second important class of elements is the class of separating twists—these are Dehn twists $T_c$ for c a separating curve. The genus of a separating twist $T_c$ that bounds a subsurface of genus h is defined as $g(c) = \min \{h, g-h\}$.

The Johnson homomorphism A fundamental tool in the study of ${\mathcal {I}}_g$ is the Johnson homomorphism, due to Johnson [10]. This is a surjective homomorphism

$$\begin{aligned} \tau {:}\,{\mathcal {I}}_g \rightarrow \wedge ^3 H_{\mathbb {Z}}/ H_{\mathbb {Z}}, \end{aligned}$$

(4)

where for convenience we define $H_A\,{:}{=}\, H_1(\Sigma _g; A)$ for some abelian group A. The embedding $H_{\mathbb {Z}}\hookrightarrow \wedge ^3 H_{\mathbb {Z}}$ is defined via

$$\begin{aligned} z \mapsto z \wedge (x_1 \wedge y_1 + \cdots + x_g \wedge y_g), \end{aligned}$$

where $\{x_1, \dots , y_g\}$ is a symplectic basis for $H_{\mathbb {Z}}$. Recall that a symplectic basis must satisfy $\langle x_i, y_i \rangle = 1$ and $\langle x_i,x_j \rangle = \langle x_i,y_j \rangle = 0$ for $i \ne j$.

We will not need to know a precise definition of $\tau$, but it will be useful to know some basic properties of $\tau$, including how to compute $\tau$ on bounding pair maps and separating twists.

Lemma 2.8

(Johnson [10])

• (1) $\tau$ is ${{\,\mathrm{Sp}\,}}(2g; {\mathbb {Z}})$-equivariant, with respect to the conjugation action on ${\mathcal {I}}_g$ and the evident action on $\wedge ^3 H_{\mathbb {Z}}/H_{\mathbb {Z}}$.

• (2) $\tau (T_c) = 0$ for any separating twist $T_c$.

• (3) Let $c\cup d$ bound a subsurface $\Sigma _{h,2}$. Choose any further subsurface $\Sigma _{h,1} \subset \Sigma _{h,2}$, and let $\{x_1, y_1, \dots , x_h, y_h\}$ be a symplectic basis for $H_1(\Sigma _{h,1};{\mathbb {Z}})$. Then

  $$\begin{aligned} \tau (T_c T_d^{-1}) = (x_1 \wedge y_1 + \cdots + x_h \wedge y_h)\wedge [c], \end{aligned}$$

  where c is oriented with $\Sigma _{h,2}$ to the left. In the case $h = 1$, if $\alpha ,\beta ,\gamma$ is a maximal chain on $\Sigma _{1,2}$, then

  $$\begin{aligned} \tau (T_cT_d^{-1}) = [\alpha ] \wedge [\beta ] \wedge [\gamma ]. \end{aligned}$$

The Johnson kernel The Johnson kernel, written ${\mathcal {K}}_g$, is the kernel of the Johnson homomorphism:

$$\begin{aligned} {\mathcal {K}}_g \,{:}{=}\, \ker (\tau ). \end{aligned}$$

A fundamental theorem of Johnson gives an alternate characterization of ${\mathcal {K}}_g$ in terms of separating twists.

Theorem 2.9

(Johnson [13]) Let ${\mathcal {T}}_g$ be the subgroup of ${\mathcal {K}}_g$ generated by all separating twists of genus at most two. Then for all $g \ge 3$,

$$\begin{aligned} {\mathcal {T}}_g = {\mathcal {K}}_g. \end{aligned}$$

3 Spin structures

In this section we introduce and study higher spin structures and their stabilizer subgroups. Section 3.1 defines higher spin structures and presents the work of Humphries–Johnson that gives a cohomological formulation of a higher spin structure. Section 3.2 discusses some cut-and-paste operations on simple closed curves and how these operations interact with higher spin structures. Section 3.3 defines spin structure stabilizer groups and some important elements of these groups. Finally Sect. 3.4 explains the connection between higher spin structures and the classical theory of spin structures as quadratic forms on vector spaces over ${\mathbb {Z}}/2{\mathbb {Z}}$.

3.1 Spin structures

Let S be a surface of genus $g \ge 0$. For simplicity, we assume in this section that S can have boundary components but not punctures; for surfaces with puncture, one can simply remove an open neighborhood of the puncture to produce a surface with boundary. Let ${\mathcal {S}}$ denote the set of isotopy classes of oriented simple closed curves on S. In keeping with standard practice, the term “curve” will often be used to refer to an isotopy class of curves. Crucially, curves are not required to be essential [see property (2) of Definition 3.1]. The following definition is due to Humphries and Johnson [9]; see Remark 3.2 for a discussion of how to reconcile their definition with the one given here.

Definition 3.1

(spin structure) A ${\mathbb {Z}}/r {\mathbb {Z}}$-valued spin structure on S is a function $\phi {:}\,{\mathcal {S}} \rightarrow {\mathbb {Z}}/r{\mathbb {Z}}$ satisfying the following two properties.

• (1) (Twist-linearity) Let $c,d \in {\mathcal {S}}$ be arbitrary. Then

  $$\begin{aligned} \phi (T_c(d)) = \phi (d) + \langle d,c \rangle \phi (c) \pmod {r}. \end{aligned}$$

• (2) (Normalization) For $\zeta$ the boundary of an embedded disk $D \subset S$, oriented with D to the left, $\phi (\zeta ) = 1$.

Remark 3.2

The definition of a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure presented in Definition 3.1 is superficially different from that given by Humphries and Johnson [9] in several respects. First, it should be noted that Humphries–Johnson study a more general notion of “twist-linear function”; only spin structures are needed in the present paper. Secondly, in Definition 3.1, simple closed curves are considered up to the equivalence relation of isotopy. This is an a priori different equivalence relation than the notion of “L-direct homotopy” defined in [9, p. 366]. The precise definition of L-directness is cumbersome, but if two simple closed curves c and d are L-directly homotopic, then they are in particular homotopic in the ordinary sense. It is well-known that homotopy and isotopy determine the same equivalence relation on simple closed curves, see e.g. [7, Proposition 1.10]. Moreover, an isotopy is an instance of an L-direct homotopy, so that these notions coincide in our setting.

Remark 3.3

In the literature, higher spin structures go by various names and have various definitions; the term “r-spin structure” is especially common. It is not a priori clear how to reconcile the definition given here with others. See Remark 3.7 for a brief discussion, or [17, Sections 2–3] for a fuller treatment.

Convention 3.4

Often we will speak of the value $\phi (c)$ where $\phi$ is some ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure and c is a curve without a specified orientation. Such a statement should be understood to mean that there is some unspecified orientation on c for which $\phi (c)$ has the stated value.

The Johnson lift Recall from the discussion in Sect. 2.2 the notion of the Johnson lift. In [9], Johnson-Humphries use the Johnson lift to give a homological formulation of a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure. The following is an amalgamation of the Remark following Theorem 2.1 and Theorem 2.5 of [9].

Theorem 3.5

(Humphries–Johnson) Let S be a surface. An element $\psi \in H^1(\textit{UTS}; {\mathbb {Z}}/r{\mathbb {Z}})$ determines a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure via

$$\begin{aligned} \alpha \mapsto \psi ({\tilde{\alpha }}), \end{aligned}$$

where $\alpha$ is a simple closed curve on S and ${\tilde{\alpha }}$ is the Johnson lift. This determines a $1-1$ correspondence

$$\begin{aligned} \{\phi \hbox { a} {\mathbb {Z}}/r{\mathbb {Z}}\hbox {-valued spin structure on} S\} \leftrightarrow \left\{ \phi \in H^1(\textit{UTS}; {\mathbb {Z}}/r{\mathbb {Z}}) \mid \phi (\zeta ) = 1\right\} . \end{aligned}$$

Remark 3.6

From the standard presentation

$$\begin{aligned} \pi _1(\textit{UT}\Sigma _g) = \left\langle a_1, b_1,\dots , a_g, b_g, \zeta \mid \prod _{i =1}^g [a_i, b_i] = \zeta ^{2-2g}\right\rangle \end{aligned}$$

and the Universal Coefficient Theorem, one sees that

$$\begin{aligned} H^1(\textit{UT}\Sigma _g; A) \cong {{\,\mathrm{Hom}\,}}(\pi _1(\textit{UT}\Sigma _g), A) \cong {{\,\mathrm{Hom}\,}}({\mathbb {Z}}^{2g}\oplus {\mathbb {Z}}/(2g-2){\mathbb {Z}}, A). \end{aligned}$$

The factor ${\mathbb {Z}}/(2g-2){\mathbb {Z}}$ in $H_1(\textit{UT}\Sigma _g;{\mathbb {Z}}) = {\mathbb {Z}}^{2g} \oplus {\mathbb {Z}}/(2g-2){\mathbb {Z}}$ is generated by the class of ${\tilde{\zeta }}$, the Johnson lift of the non-essential curve $\zeta$. In the case $A = {\mathbb {Z}}/r {\mathbb {Z}}$, it follows that there exists a spin structure if and only if $r \mid (2g-2)$.

Remark 3.7

Via covering space theory, ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structures on $\Sigma _g$ are in correspondence with cyclic r-fold coverings $\widetilde{\textit{UT}\Sigma _g} \rightarrow UT \Sigma _g$ that restrict to connected coverings of the fiber $S^1$. In the setting of linear systems on toric surfaces, such coverings arise from the presence of roots of the canonical line bundle of the generic fiber. See Proposition 10.2 and the references mentioned therein for more details.

An important consequence of Theorem 3.5 is the fact that ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structures satisfy a property known as the homological coherence criterion. This follows by combining Theorem 3.5 with [9, Lemma 2.4].

Proposition 3.8

(Homological coherence criterion) Let $\phi$ be a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure on a surface S, and let $S' \subset S$ be a subsurface with Euler characteristic $\chi (S') = m$. Suppose $\partial (S') = c_1 \cup \cdots \cup c_k$, and all $c_i$ are oriented so that $S'$ is to the left. Then $\sum \phi (c_i) = m$.

Theorem 3.5 shows that ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structures are determined by a finite amount of data. In the sequel it will be useful to have an explicit criterion for the equality of two ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structures. The following appears as [9, Corollary 2.6].

Theorem 3.9

(Humphries–Johnson) Let S be a surface of genus $g \ge 0$. Let ${\mathcal {B}} = \{\gamma _1, \dots , \gamma _{k}\}$ be a set of oriented simple closed curves such that the set $\{[\gamma _1], \dots , [\gamma _k]\}$ forms a basis for $H_1(\Sigma _g;{\mathbb {Z}})$. Suppose $\phi$ and $\psi$ are ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structures on S. Then $\phi = \psi$ if and only if $\phi (\gamma _i) = \psi (\gamma _i)$ for each $\gamma _i \in {\mathcal {B}}$.

3.2 Operations on curves

In what follows, we will make use of two procedures for constructing new simple closed curves from old. Here, we define these operations and collect some facts about how they interact with spin structures.

Definition 3.10

(Smoothing, curve sum) Let ${\mathcal {C}} = \{c_1, \ldots , c_n\}$ be a collection of oriented embedded simple closed curves on a surface S. Suppose that all intersections between elements of ${\mathcal {C}}$ are transverse. The smoothing of ${\mathcal {C}}$ is the embedded multicurve obtained from ${\mathcal {C}}$ by smoothly resolving all intersections in the unique orientation-preserving way. See Fig. 3.

Fig. 3

The smoothing operation

Now suppose $\alpha$ and $\beta$ are oriented simple closed curves. For natural numbers m, n, define the curve sum$m \alpha + n \beta$ as the smoothing of m parallel copies of $\alpha$ with n copies of $\beta$. In case $m < 0$ or $n < 0$, the curve sum $m \alpha + n \beta$ can be defined as before, with the orientation on $\alpha$ (resp. $\beta$) reversed if $m < 0$ (resp. $n < 0$). See Fig. 4.

Fig. 4

The curve-arc sum operation

By choosing arbitrary representatives in minimal position, both of these operations are well-defined on the level of isotopy classes.

Lemma 3.11

Let $\alpha , \beta$ be oriented simple closed curves in minimal position, and let $\phi$ be a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure. Then for any integers m, n,

$$\begin{aligned} \phi (m\alpha + n \beta ) = m \phi (\alpha ) + n \phi (\beta ). \end{aligned}$$

If in addition, $i(\alpha , \beta ) = 1$ and $\gcd (m,n) =1$, then $m \alpha + n \beta$ has a single component.

Proof

The first assertion follows directly from the identification of $\phi$ with an element of $H^1(\textit{UTS}; {\mathbb {Z}}/r{\mathbb {Z}})$ given in Theorem 3.5, while the second is straightforward to verify. $\square$

Definition 3.12

(Curve-arc sum) Let $\alpha$ and $\beta$ be disjoint oriented simple closed curves on S, and let $\epsilon$ be an arc connecting $\alpha$ to $\beta$ whose interior is disjoint from $\alpha \cup \beta$. A regular neighborhood $\nu$ of $\alpha \cup \epsilon \cup \beta$ is homeomorphic to $\Sigma _{0,3}$. Two of the boundary components of $\nu$ are homotopic to $\alpha$ and $\beta$, respectively. The curve-arc-sum$\alpha +_{\epsilon } \beta$ is the third boundary component of $\nu$. Again, the curve-arc sum descends to the level of isotopy classes.

Lemma 3.13

Let $\alpha , \beta , \epsilon , \nu$ be as above. Orient $\alpha , \beta$ so that $\epsilon$ connects the left sides of $\alpha , \beta$, and orient $\alpha +_{\epsilon } \beta$ so that the subsurface $\nu$ is to the right. Then for $\phi$ a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure,

$$\begin{aligned} \phi (\alpha +_{\epsilon }\beta ) = \phi (\alpha ) + \phi (\beta ) + 1. \end{aligned}$$

In addition, on the level of homology, $[\alpha +_{\epsilon } \beta ] = [\alpha ] + [\beta ]$.

Proof

Observe that $\chi (\nu ) = -1$. By the homological coherence criterion (Proposition 3.8),

$$\begin{aligned} -1 = \phi (\alpha ) + \phi (\beta ) + \phi (-(\alpha +_\epsilon \beta )), \end{aligned}$$

where $-(\alpha +_\epsilon \beta )$ denotes the curve $\alpha +_\epsilon \beta$ with orientation opposite to that specified above. By the case $(m,n) = (-1,0)$ of Lemma 3.11, it follows that $\phi (-(\alpha +_\epsilon \beta )) = - \phi (\alpha +_\epsilon \beta )$, from which the first claim follows. The second claim is an immediate consequence of the orientation conventions. $\square$

3.3 The group ${{\,\mathrm{Mod}\,}}(S)[\phi ]$; first examples of elements

For any surface S, there is an obvious (left) action of ${{\,\mathrm{Mod}\,}}(S)$ on the set of ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structures: for $f \in {{\,\mathrm{Mod}\,}}(S)$ and $c \in {\mathcal {S}}$, define $(f\cdot \phi )(c) = \phi (f^{-1}(c))$. Similarly, if $f{:}\,S \rightarrow S'$ is a diffeomorphism and $\phi$ is a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure on $S'$, there is a pullback $f^*(\phi )$ defined on S via $(f^*\phi )(c) = \phi (f(c))$.

Definition 3.14

(Stabilizer subgroup) Let $\phi$ be a spin structure on a surface S. The stabilizer subgroup of $\phi$, written ${{\,\mathrm{Mod}\,}}(S)[\phi ]$, is defined as

$$\begin{aligned} {{\,\mathrm{Mod}\,}}(S)[\phi ] = \left\{ f\in {{\,\mathrm{Mod}\,}}(S) \mid (f \cdot \phi ) = \phi \right\} . \end{aligned}$$

Let $\phi$ be a ${\mathbb {Z}}/r {\mathbb {Z}}$-valued spin structure on a surface S. Below we discuss some fundamental examples of elements in ${{\,\mathrm{Mod}\,}}(S)[\phi ]$.

Dehn twist powers and admissible twists The twist-linearity formula of Definition 3.1 immediately implies the following characterization of Dehn twists in ${{\,\mathrm{Mod}\,}}(S)[\phi ]$.

Lemma 3.15

Let c be a simple closed curve on S. If c is separating, then $T_c \in {{\,\mathrm{Mod}\,}}(S)[\phi ]$. If c is nonseparating, then $T_c^k \in {{\,\mathrm{Mod}\,}}(S)[\phi ]$ if and only if $k \phi (c) \equiv 0 \pmod r$. In particular, for c nonseparating, $T_c \in {{\,\mathrm{Mod}\,}}(S)[\phi ]$ if and only if $\phi (c) = 0$.

Definition 3.16

(Admissible) A nonseparating curve c with $\phi (c) = 0$ is called an admissible curve. The associated element $T_c \in {{\,\mathrm{Mod}\,}}(S)[\phi ]$ is called an admissible twist. The group generated by all admissible twists is written ${\mathcal {T}}_\phi$, and is called the admissible subgroup.

Fundamental multitwists Let $P\cong \Sigma _{0,3}$ be a pair of pants with boundary curves $\alpha ,\beta ,\gamma$. Suppose that $\phi (\alpha ) = a, \phi (\beta ) = b,$ and that $\phi (\gamma ) = c$, with all curves oriented so that P lies to the left. By the homological coherence property, $a + b + c = -1$.

Definition 3.17

Let P and $\phi$ be as above. A $\phi$-bounding multitwist associated to P, denoted $T_P(x,y,z)$, is given by

$$\begin{aligned} T_P(x,y,z) = T_\alpha ^{x}T_\beta ^y T_\gamma ^z \end{aligned}$$

for any choice of integers x, y, z such that $T_P \in {{\,\mathrm{Mod}\,}}(S)[\phi ]$.

By the above, $T_P(r, r, r)$ is a $\phi$-bounding multitwist for any P and $\phi$, but for special values of a, b, c, there are much simpler examples.

Lemma 3.18

Let P be as above, and suppose that $b = -a$, so that $c = -1$. Then $T_P(1,-1, b) = T_\alpha T_\beta ^{-1} T_\gamma ^{b}$ is a $\phi$-bounding multitwist. The element $T_P(1,-1,b)$ is called a fundamental multitwist for P and is denoted $T_P$.

Proof

Let d be any curve on S; we must show that $\phi (d) = \phi (T_\alpha T_\beta ^{-1} T_\gamma ^b(d))$. As $\alpha , \beta , \gamma$ are all disjoint, the twist-linearity property, in combination with the fact that $[\alpha + \beta + \gamma ] = 0$ in $H_1(S)$, gives

$$\begin{aligned} \phi (T_\alpha T_\beta ^{-1} T_\gamma ^b(d))&= \phi (d) + \langle d,\alpha \rangle a - \langle d,\beta \rangle b - \langle d, \gamma \rangle b\\&= \phi (d) - \langle d, \alpha + \beta + \gamma \rangle b\\&= \phi (d). \end{aligned}$$

$\square$

Remark 3.19

Of course, if $T_P(1,-1,b)$ is a fundamental multitwist, then so is $T_P(1,-1, b+ k r)$ for any $k \in {\mathbb {Z}}$. An important special case is when $\phi (\alpha ) = \phi (\beta ) = 0$. Then $T_\alpha T_\beta ^{-1}$ is a fundamental multitwist.

3.4 “Classical” spin structures

Spin structures in the sense of Definition 3.1 generalize the more familiar notion of a “classical” spin structure. In our setting, a classical spin structure is a spin structure valued in ${\mathbb {Z}}/2{\mathbb {Z}}$. We pause here to briefly review the theory of classical spin structures and the connection with our definition. These results, especially the theory of the Arf invariant, will play a crucial role in the analysis of ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structures for r even to be begun in Proposition 4.9 and Corollary 4.10, and returned to in Sect. 6.

Let V be a vector space over the field ${\mathbb {Z}}/2{\mathbb {Z}}$ equipped with a nondegenerate symplectic pairing $\langle \cdot ,\cdot \rangle$ (i.e. a nondegenerate bilinear pairing satisfying $\langle x,x \rangle = 0$ for all $x \in V$). The motivating example is $V = H_1(\Sigma _g; {\mathbb {Z}}/2{\mathbb {Z}})$ with the intersection pairing. A ${\mathbb {Z}}/2{\mathbb {Z}}$quadratic form relative to$\langle \cdot , \cdot \rangle$ is a function $q{:}\,V \rightarrow {\mathbb {Z}}/2{\mathbb {Z}}$ such that for any $x,y \in V$, the equation

$$\begin{aligned} q(x+y) = q(x) + q(y) + \langle x,y \rangle \end{aligned}$$

(5)

holds.

Let ${\mathcal {B}} = \{x_1, y_1, \dots , x_g, y_g\}$ be a symplectic basis for V. It is clear that q is determined by its values on ${\mathcal {B}}$. Define $Q(V, \langle \cdot , \cdot \rangle )$ as the set of ${\mathbb {Z}}/2{\mathbb {Z}}$ quadratic forms on V relative to $\langle \cdot , \cdot \rangle$; then a choice of ${\mathcal {B}}$ provides a bijection

$$\begin{aligned} Q(V, \langle \cdot ,\cdot \rangle ) \cong ({\mathbb {Z}}/2{\mathbb {Z}})^{2g}. \end{aligned}$$

There is an evident action of the group ${{\,\mathrm{Sp}\,}}(V, \langle \cdot , \cdot \rangle )$ of $\langle \cdot ,\cdot \rangle$-preserving automorphisms on $Q(V, \langle \cdot ,\cdot \rangle )$.

To understand the set of orbits, we introduce the Arf invariant. The Arf invariant of q is the element of ${\mathbb {Z}}/2{\mathbb {Z}}$ defined by the following formula:

$$\begin{aligned} {{\,\mathrm{Arf}\,}}(q) \,{:}{=}\, \sum _{i = 1}^g q(x_i)q(y_i). \end{aligned}$$

q is said to be even or odd according to whether ${{\,\mathrm{Arf}\,}}(q) = 0,1$ respectively; in this way we will speak of the parity of a spin structure. The following records some well-known properties of the Arf invariant.

Lemma 3.20

Let $(V, \langle \cdot , \cdot \rangle )$ be a symplectic vector space over ${\mathbb {Z}}/2{\mathbb {Z}}$, and let $q,q' \in Q(V, \langle \cdot ,\cdot \rangle )$ be quadratic forms.

• (1) ${{\,\mathrm{Arf}\,}}(q)$ is well-defined independently of the choice of symplectic basis,

• (2) q and $q'$ are in the same orbit of ${{\,\mathrm{Sp}\,}}(V, \langle \cdot , \cdot \rangle )$ if and only if ${{\,\mathrm{Arf}\,}}(q) = {{\,\mathrm{Arf}\,}}(q')$.

Suppose now that $\phi$ is a ${\mathbb {Z}}/2d{\mathbb {Z}}$-valued spin structure in the sense of Definition 3.1. The reduction ${\mathbb {Z}}/2d{\mathbb {Z}}\rightarrow {\mathbb {Z}}/2{\mathbb {Z}}$ associates to $\phi$ an underlying ${\mathbb {Z}}/2{\mathbb {Z}}$-valued spin structure which we denote ${\bar{\phi }}$. A priori, ${\bar{\phi }}$ is defined on the set ${\mathcal {S}}$ of isotopy classes of oriented curves on $\Sigma _g$. It follows from [11, Theorem 1A] that ${\bar{\phi }}$ factors through the map $[\cdot ]{:}\,{\mathcal {S}} \rightarrow H_1(\Sigma _g;{\mathbb {Z}}/2{\mathbb {Z}})$. The induced map

$$\begin{aligned} \bar{{\bar{\phi }}}{:}\,H_1(\Sigma _g;{\mathbb {Z}}/2{\mathbb {Z}})\rightarrow {\mathbb {Z}}/2{\mathbb {Z}}\end{aligned}$$

is not quite a classical spin structure, but it follows from [11, Theorem 1A] that the function

$$\begin{aligned} q_\phi \,{:}{=}\, \bar{{\bar{\phi }}} + 1 \end{aligned}$$

(6)

does determine a classical spin structure.

In the remainder of this paper we will exclusively use the term “spin structure” in the sense of Definition 3.1. The reader versed in classical spin structures should be aware that certain formulas appear different in this setting. For instance, a Dehn twist about some nonseparating curve c preserves a ${\mathbb {Z}}/2{\mathbb {Z}}$-valued spin structure $\phi$ if and only if $\phi (c) = 0$, whereas a transvection about some nonzero $v\in V$ preserves a quadratic form q if and only if $q(v) = 1$. Likewise, if $\phi$ is a ${\mathbb {Z}}/2d{\mathbb {Z}}$-valued spin structure, the formula for the Arf invariant ${{\,\mathrm{Arf}\,}}(\phi )$ of the underlying classical spin structure is given by

$$\begin{aligned} {{\,\mathrm{Arf}\,}}(\phi ) = \sum _{i = 1}^g ({\bar{\phi }}(x_i) + 1)({\bar{\phi }}(y_i) + 1) \pmod 2. \end{aligned}$$

(7)

4 The action of the mapping class group on spin structures

In what follows, we will need to understand the action of ${{\,\mathrm{Mod}\,}}(\Sigma _g)$ on the set of ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structures. Following the discussion in Sect. 3.4, when r is even, the Arf invariant shows there are at least two orbits of ${{\,\mathrm{Mod}\,}}(S)$ on the set of ${\mathbb {Z}}/r {\mathbb {Z}}$-valued spin structures, but it is not clear what happens for odd r, nor whether there are further invariants leading to more orbits. The goal of this section is to give a complete description of this action. In the case of r odd, the mapping class group action on the set of ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structures is described in Proposition 4.2, and for r even it is described in Proposition 4.9. Both results can be understood as asserting that there are no “higher Arf invariants”.

4.1 Odd r

In the case of r odd, we will need to consider surfaces with multiple boundary components. Before formulating the results, we define the notion of the signature of a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure.

Definition 4.1

(Signature of a${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure) Let S be a surface equipped with a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure $\phi$. Enumerate the boundary components as $\Delta _1, \dots , \Delta _n$, each one oriented so that S is to the left. The signature ofSrel$\phi$ is defined as the n-tuple of values ${{\,\mathrm{sig}\,}}(S, \phi ) = (\phi (\Delta _1), \dots , \phi (\Delta _n))$. We will also speak of the signature of an individual $\Delta _k$, defined as $\phi (\Delta _k)$.

Proposition 4.2

Fix an odd integer r. Let S be a surface, and let $\phi$ and $\psi$ be ${\mathbb {Z}}/r {\mathbb {Z}}$-valued spin structures on S satisfying ${{\,\mathrm{sig}\,}}(\phi ) = {{\,\mathrm{sig}\,}}(\psi )$. Suppose that either $g(S) \ne 1$ or else $g = 1$ and there is at least one boundary component with signature $\phi (c_1) = \psi (c_1) = k$ for some k such that $k+1$ generates ${\mathbb {Z}}/r {\mathbb {Z}}$. Then there is a mapping class $f \in {{\,\mathrm{Mod}\,}}(S)$ such that $f^*(\psi ) = \phi$.

Proof

The proof is by induction on the genus g(S). If $g(S) = 0$, then every curve c on S is separating, so that the homological coherence criterion (Proposition 3.8) implies that $\phi (c)$ and $\psi (c)$ can be computed just from the signature. In this case, it follows that in fact $\phi = \psi$.

For $g(S) \ge 1$, let $\alpha _0,\beta _0$ be curves on S satisfying $i(\alpha _0,\beta _0) = 1$. Choose nonzero integers $a,b \in {\mathbb {Z}}$ such that $a \equiv \phi (\alpha _0)$ and $b \equiv \phi (\beta _0) \pmod r$. Let $d = \gcd (a,b)$, and define $x = a/d, y=b/d$; by construction, x, y are coprime. Define the curve $\alpha _1 = y\alpha _0 -x \beta _0$ in the sense of Definition 3.10. By Lemma 3.11, $\phi (\alpha _1) = 0$.

Choose any curve $\gamma _0$ satisfying $i(\alpha _1, \gamma _0) = 1$. We claim there exists some separating oriented curve c on S that is disjoint from $\gamma _0 \cup \alpha _1$ and such that $\phi (c) = k$ for k such that $k+1$ generates ${\mathbb {Z}}/r {\mathbb {Z}}$. In the case $g(S) = 1$ this is true by hypothesis, while for $g(S) \ge 2$, the curve c can be taken to be the neighborhood of some subsurface $T \subset S$ with $T \cong \Sigma _{1,1}$ and T disjoint from $\alpha _1 \cup \gamma _0$. In this case, orient c so that T lies to the right. By the homological coherence property, such a c satisfies $\phi (c) = 1$, and since r is odd, the claim follows.

Either c is isotopic to a boundary component of S and is oriented with S lying to the right, or else (by the change-of-coordinates principle), there exists an arc $\epsilon _0$ from the left side of $\gamma _0$ to the left side of c that is disjoint from $\alpha _1$. In the former case, there exists an arc $\epsilon _0$ from the right side of $\gamma _0$ to the right side of c that is disjoint from $\alpha _1$. Via Lemma 3.13, the curve-arc sum $\gamma _1 = \gamma _0 +_{\epsilon _0} c$ satisfies $\phi (\gamma _1) = \phi (\gamma _0) - (k+1)$ in the former case, and $\phi (\gamma _1) = \phi (\gamma _0) + (k+1)$ in the latter case. Since the curve c is null-homologous, there is an equality $[\gamma _1] = [\gamma _0]$. A further appeal to the change-of-coordinates principle shows that there is another arc $\epsilon _1$ from the left side of $\gamma _1$ to the left of c, again disjoint from $\alpha _1$. This process can therefore be repeated indefinitely, giving rise to curves $\gamma _m$ satisfying $\phi (\gamma _m) = \phi (\gamma _0) + m(k+1)$. By hypothesis, $k+1 \in {\mathbb {Z}}/r {\mathbb {Z}}$ is a generator, so that $\phi (\gamma _m) = 0$ for some m. Set $\beta _1 = \gamma _m$ for such an m. By construction, $i(\alpha _1, \beta _1) = 1$.

Likewise, construct curves $\alpha _1', \beta _1'$ satisfying $i(\alpha _1', \beta _1') = 1$ and $\psi (\alpha _1') = \psi (\beta _1') = 0$. Take (open) regular neighborhoods $T_1$ and $T_1'$ of $\alpha _1\cup \beta _1$ and $\alpha _1' \cup \beta _1'$, respectively. There is a diffeomorphism $f_1{:}\,T_1 \rightarrow T_1'$ for which $f_1(\alpha _1) = \alpha _1'$ and $f_1(\beta _1) = \beta _1'$. Define $c_1 = \partial \overline{T_1}$ and $c_1' = \partial \overline{T_1'}$. Then $\phi (c_1) = 1$ when $c_1$ is oriented with $T_1$ on the right, and similarly for $c_1'$. The curve $c_1$ is therefore a boundary component of $S {\setminus } T_1$ with signature $\phi (c_1) = 1$, and likewise for $c_1'$. This shows that the inductive hypothesis is satisfied, and so there exists a diffeomorphism $f_2{:}\,S {\setminus } T_1 \rightarrow S{\setminus } T_1'$ taking $c_1$ to $c_1'$ and fixing each remaining mutual boundary component, such that

$$\begin{aligned} f_2^*(\psi \mid _{S {\setminus } T_1'}) = \phi \mid _{S {\setminus } T_1.} \end{aligned}$$

The diffeomorphisms $f_1$ and $f_2$ can be chosen in such a way as to extend to a diffeomorphism $f{:}\,S \rightarrow S$. Let ${\mathcal {B}} = \{\alpha _1, \beta _1, \dots , \alpha _g, \beta _g\}$ be a geometric symplectic basis for S, with $\alpha _1, \beta _1$ the same curves as above. Necessarily $\alpha _k, \beta _k$ are curves on $S {\setminus } T_1$ for $k \ge 2$. By construction, the spin structures $\phi$ and $f^*(\psi )$ take the same values on each element of ${\mathcal {B}}$, and ${{\,\mathrm{sig}\,}}(S, \phi ) = {{\,\mathrm{sig}\,}}(S, f^*(\psi ))$. It then follows from Proposition 3.9 that $\phi = f^*(\psi )$ as claimed. $\square$

Proposition 4.2 has several corollaries that will be used extensively in the remainder of the paper. These play the role of a change-of-coordinates principle for surfaces in the presence of a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure. The first of these was established in the second paragraph of the proof of Proposition 4.2. We remark that the assumption that r is odd played no role in the argument.

Corollary 4.3

Let r be an integer and let $\phi$ be a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure on a surface S. Let $S' \subset S$ be a subsurface of genus $h \ge 1$. Then there is some admissible curve $a \subset S'$ that is not parallel to a boundary component.

This in turn leads to another useful result that will allow us to construct curves with prescribed intersection properties and arbitrary $\phi$-values.

Corollary 4.4

Let r be an integer and let $\phi$ be a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure on a surface S. Let $a, c_1, \dots , c_k$ be a collection of simple closed curves. Assume that there is some connected subsurface $T \subset S$ of positive genus disjoint from $a, c_1, \dots , c_k$, and that there is an arc $\epsilon$ connecting a to $\partial T$ that is disjoint from all $c_i$. Then for $\ell \in {\mathbb {Z}}/r{\mathbb {Z}}$ arbitrary, there is a simple closed curve $a_\ell$ for which $i(a_\ell , c_i) = i(a,c_i)$ for $i = 1, \dots , k$, and for which $\phi (a_\ell ) = \ell$.

Proof

By Corollary 4.3, there exists an admissible curve $b \subset T$ that is not boundary-parallel. The arc $\epsilon$ can be concatenated with an arc joining $\partial T$ to b; denote this extended arc by $\epsilon '$. Set $\ell _0 = \phi (a)$ (where a is oriented with $\epsilon '$ lying to the left), and define $a_{\ell _0}\,{:}{=}\, a$. Define $a_{\ell _0+1} \,{:}{=}\, a_{\ell _0} +_{\epsilon '} b$. By Lemma 3.13, $\phi (a_{\ell _0+1}) = \phi (a_{\ell _0}) + 1 = \ell _0 + 1$.

To see that $i(a_{\ell _0+1}, c_i) = i(a, c_i)$, we appeal to the bigon criterion of [7, Proposition 1.7]. Choose representative curves for $a, c_1, \dots , c_k$, pairwise in minimal position. The bigon criterion asserts that $a, c_i$ are in minimal position if and only if the configuration $a \cup c_i$ does not bound any bigons, i.e. an embedded disk whose boundary is the union of an arc of a and an arc of $c_i$ meeting in exactly two points. The curve-arc sum $a_{\ell _0 +1}$ meets each $c_i$ in exactly the same set of points as $a_{\ell _0}$. To conclude, it thus suffices to see that no bigons were introduced by the summing procedure. The only arc of $a_{\ell _0+1}$ that is not also an arc of $a_{\ell _0}$ is the one along which the summing procedure is performed; denote the original arc of $\alpha _{\ell _0}$ by $\alpha$ and the modified arc by $\alpha '$. Suppose that there is an arc $\gamma$ of $c_i$ such that $\alpha ' \cup \gamma$ bounds a bigon. As $\alpha ' = \alpha +_{\epsilon '} b$, it must be the case that the curve $\alpha \cup \gamma$ is isotopic to b. But by assumption, $b \subset T$ is not boundary-parallel, so this cannot happen.

To construct $a_\ell$ for $\ell \in {\mathbb {Z}}/r{\mathbb {Z}}$ arbitrary, one simply repeats the above construction, producing, for any $t \ge 0$, a curve $a_{\ell _0+t}$ with the same intersection properties as $a_{\ell _0}$ and satisfying $\phi (a_{\ell _0 + t}) = \ell _0 + t$. $\square$

For the remaining corollaries of Proposition 4.2, we re-instate the requirement that r be odd.

Corollary 4.5

Let r be an odd integer and let $\phi$ be a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure on a surface S. Let $S' \subset S$ be a subsurface of genus $h \ge 1$, and suppose that if $h = 1$, then $S'$ includes some boundary component of signature k such that $k+1$ generates ${\mathbb {Z}}/r {\mathbb {Z}}$.

• (1) For all $x \in {\mathbb {Z}}/r {\mathbb {Z}}$, there exists some nonseparating curve c supported on $S'$ satisfying $\phi (c) = x$,

• (2) For any 2h-tuple $(i_1, j_1, \dots , i_h, j_h)$ of elements of ${\mathbb {Z}}/ r {\mathbb {Z}}$, there is some geometric symplectic basis ${\mathcal {B}} = \{a_1, b_1, \dots , a_h, b_h\}$ for $S'$ with $\phi (a_\ell ) = i_\ell$ and $\phi (b_\ell ) = j_\ell$ for all $1 \le \ell \le h$,

• (3) For any 2h-tuple $(k_1, \dots , k_{2h})$ of elements of ${\mathbb {Z}}/r{\mathbb {Z}}$, there is some chain $(a_1, \dots , a_{2h})$ of curves on $S'$ such that $\phi (a_\ell ) = k_\ell$ for all $1 \le \ell \le 2h$.

Proof

Certainly (1) follows from (2). To establish (3), choose any geometric symplectic basis ${\mathcal {B}} = \{a_\ell ', b_\ell '\}$ on $S'$. There is some spin structure $\psi$ on $S'$ for which $\psi (a_\ell ') = i_\ell$ and $\psi (b_\ell ') = j_\ell$. By Proposition 4.2, there is a diffeomorphism f of $S'$ such that $f^*(\psi ) = \phi$. Then ${\mathcal {B}} = f^{-1}({\mathcal {B}}')$ has the required properties.

We will deduce (3) from (2). Given the 2h-tuple $(k_1, \dots , k_{2h})$, define a 2h-tuple $(i_1, j_1, \dots , i_h, j_h)$ as follows: set $i_\ell =1 -\ell + \sum _{t = 1}^\ell k_{2t - 1}$, and set $j_\ell = k_{2\ell }$. By (2), there exists a geometric symplectic basis ${\mathcal {B}} = \{c_\ell , d_\ell \}$ on $S'$ whose $\phi$-values realize the tuple $(i_1, j_1, \dots , i_h, j_h)$. Any geometric symplectic basis can be “completed” into a chain as follows: for $\ell = 1, \dots , h-1$, let $f_\ell$ be a simple closed curve satisfying $i(f_\ell , d_{\ell }) = i(f_\ell , d_{\ell +1}) = 1$ and $i(f_\ell ,x) = 0$ for all other elements $x \in {\mathcal {B}}$. As ${\mathcal {B}}$ is a geometric symplectic basis, this imposes the homological relation $[f_\ell ] = [c_{\ell +1}]-[c_{\ell }]$, and the intersection conditions imposed on the set of curves $\{f_\ell \}$ imply that this homology is realized geometrically: $c_\ell \cup f_\ell \cup c_{\ell +1}$ must bound a pair of pants $P_\ell$ for each $\ell = 1, \dots , h-1$. The orientations can be arranged so that $P_\ell$ lies to the right of $c_\ell$ and $f_\ell$ and to the left of $c_{\ell +1}$.

Applying the homological coherence property to each $P_\ell$, it follows that $\phi (f_\ell ) = k_{2 \ell + 1}$. By construction, the curves $c_1, d_1, f_1, d_2, f_2, d_3,\dots , f_{h-1}, d_h$ form a chain of length 2h; denote this chain by C. By construction, $\phi (c_1) = i_1 = k_1$, and $\phi (d_\ell ) = k_{2 \ell }$. Altogether, this shows that C has the required properties. $\square$

4.2 Even r

Following the discussion in Sect. 3.4, we see that the Arf invariant distinguishes at least two orbits of ${{\,\mathrm{Mod}\,}}(\Sigma _g)$ on the set of ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structures. To see that there are exactly two orbits, in Definition 4.6 we formulate two “model” ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structures $\phi ^{{\mathcal {B}}}_{even}$ and $\phi ^{{\mathcal {B}}}_{odd}$ of prescribed Arf invariant, and in Proposition 4.9 we show that every ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure is equivalent to one of $\phi ^{{\mathcal {B}}}_{even}$ or $\phi ^{{\mathcal {B}}}_{odd}$. We restrict attention here to the case where the surface S has at most one boundary component. The general setting of multiple boundary components introduces considerable subtlety owing to the failure for the intersection pairing to determine a symplectic form, and our results require only the case of at most one boundary component.

Definition 4.6

Let S be a surface of genus $g \ge 1$ with at most one boundary component. Fix a geometric symplectic basis ${\mathcal {B}} = \{\alpha _1, \beta _1, \dots , \alpha _g, \beta _g\}$. Define $\phi _{even}^{{\mathcal {B}}}$ and $\phi _{odd}^{{\mathcal {B}}}$ as the ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structures such that $\phi _{even}^{\mathcal B}(\gamma ) = \phi _{odd}^{{\mathcal {B}}}(\gamma )= 0$ for all $\gamma \in {\mathcal {B}} {\setminus } \{\beta _g\}$, and where $\phi _{even}^{\mathcal B}(\beta _g)$ and $\phi _{odd}^{{\mathcal {B}}}(\beta _g)$ are chosen to be 0 or 1 as necessary so that ${{\,\mathrm{Arf}\,}}(\phi _{even}^{{\mathcal {B}}}) = 0$ and ${{\,\mathrm{Arf}\,}}(\phi _{odd}^{{\mathcal {B}}}) = 1$.

In spite of the evident dependence on geometric symplectic basis, as ${\mathcal {B}}$ ranges over the set of all geometric symplectic bases, the elements $\phi ^{{\mathcal {B}}}_{odd}$ lie in a single orbit of ${{\,\mathrm{Mod}\,}}(S)$ (and the same is also true of $\phi _{even}^{{\mathcal {B}}}$). The following is immediate via the change-of-coordinates principle.

Lemma 4.7

Let ${\mathcal {B}}$ and ${\mathcal {B}}'$ be geometric symplectic bases. Then there is a diffeomorphism $f{:}\,S \rightarrow S$ such that $f({\mathcal {B}}) = {\mathcal {B}}'$. Consequently, $f^*(\phi _{even}^{{\mathcal {B}}'}) = \phi _{even}^{{\mathcal {B}}}$ and $f^*(\phi _{odd}^{{\mathcal {B}}'}) = \phi _{odd}^{{\mathcal {B}}}$.

Definition 4.8

Let S be a surface of genus $g \ge 1$ with at most one boundary component endowed with a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure $\phi$. We say that $\phi$ is even if there is a geometric symplectic basis ${\mathcal {B}}$ such that $\phi = \phi _{even}^{{\mathcal {B}}}$, and we say that $\phi$ is odd if $\phi = \phi _{odd}^{{\mathcal {B}}}$.

Proposition 4.9

Fix an even integer r. Let S be a surface of genus $g \ge 2$ with at most one boundary component. Let $\phi$ be a ${\mathbb {Z}}/r {\mathbb {Z}}$-valued spin structure on S. Then in the sense of Definition 4.8, either $\phi$ is even, or else $\phi$ is odd.

Proof

The argument makes use of the techniques of the proof of Proposition 4.2. Let ${\mathcal {B}} = \{\alpha _1, \beta _1, \dots , \alpha _g, \beta _g\}$ be a geometric symplectic basis, and let $S_i$ denote the genus-1 subsurface determined by $\alpha _i, \beta _i$; define $D_i$ as the boundary curve of $S_i$. Exactly as in Proposition 4.2, each pair $\alpha _i, \beta _i$ can be replaced by new curves $\alpha _i', \beta _i'$ supported on $S_i$ and satisfying $i(\alpha _i', \beta _i') = 1$, such that $\alpha _i'$ is admissible. Denote the corresponding geometric symplectic basis by ${\mathcal {B}}'$. For an arc $\epsilon$ connecting $\beta _1'$ to $D_2$ and disjoint from all other $D_i$, the curve-arc sum $\beta _1' +_\epsilon D_2$ satisfies $\phi (\beta _1'+_\epsilon D_2) = \phi (\beta _1') + 2$. By repeatedly performing this curve-arc sum using an arc $\epsilon$ disjoint from ${\mathcal {B}}' {\setminus } \{\beta _2'\}$ (as in Proposition 4.2), $\beta _2'$ can be replaced with a curve $\beta _2''$ such that $\phi (\beta _2'') = 0 \text{ or } 1$. By performing an analogous operation on all $\beta _i'$, one obtains a geometric symplectic basis ${\mathcal {B}}'' = \{\alpha _1', \beta _1'', \dots , \alpha _g', \beta _g''\}$ such that $\phi (\alpha _i') = 0$ and $\phi (\beta _i'') = 0 \text{ or } 1$.

It remains to further alter each $\beta _1'', \dots , \beta _{g-1}''$ so that $\phi (\beta _i'') = 0$ in this range. For $1 \le i \le g-1$, let $\gamma _i$ be a collection of disjoint curves such that $\beta _1, \gamma _1, \dots , \beta _{g-1}, \gamma _{g-1}, \beta _g$ forms a chain of length $2g-1$, and such that each $\gamma _i$ is disjoint from all $\alpha _j'$. Then necessarily $\alpha _i, \gamma _i, \alpha _{i+1}$ forms a pair of pants, and so $\phi (\gamma _i) = -1$. If $\phi (\beta _1'') = 1$, then $\phi (T_{\gamma _1}(\beta _1'')) = 0$. Replace $\beta _1'', \beta _2''$ by $T_{\gamma _1}(\beta _1''), T_{\gamma _1}(\beta _2'')$, respectively. Repeat, applying $T_{\gamma _2}^k$ to $T_{\gamma _1}({\mathcal {B}}'')$ for k such that $\phi (T_{\gamma _2}^k T_{\gamma _1}(\beta _2'')) = 0$. Proceed in this way, taking each $\beta _i''$ for $i \le g-1$ to some $\beta _i'''$ with $\phi (\beta _i''') = 0$. At the end, the geometric symplectic basis ${\mathcal {B}}''' = \{\alpha _1', \beta _1''', \dots , \alpha _g', \beta _g'''\}$ will satisfy $\phi (\gamma ) = 0$ for all $\gamma \in {\mathcal {B}}'''$ except possibly $\gamma = \beta _g'''$. By repeating the curve-arc summing procedure, $\beta _g'''$ can be altered to satisfy $\psi (\beta _g''') = 0 \text{ or } 1$ as required. Define $\widetilde{{\mathcal {B}}}$ to be this geometric symplectic basis. Applying Theorem 3.9, we see that $\phi = \phi _{even}^{\widetilde{{\mathcal {B}}}} \text{ or } \phi _{odd}^{\widetilde{{\mathcal {B}}}}$ as required. $\square$

There is an analogue of Corollary 4.5 for r even, although the Arf invariant provides an obstruction that was not present in the case of odd r.

Corollary 4.10

Let r be an even integer, and let $S' \subset S$ be a subsurface of genus $h \ge 2$ endowed with a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure $\phi$. Then the following assertions hold:

• (1) For all $x \in {\mathbb {Z}}/r {\mathbb {Z}}$, there exists some nonseparating curve c supported on $S'$ satisfying $\phi (c) = x$.

• (2) For a given 2h-tuple $(i_1, j_1, \dots , i_h, j_h)$ of elements of ${\mathbb {Z}}/ r {\mathbb {Z}}$, there is some geometric symplectic basis ${\mathcal {B}} = \{a_1, b_1, \dots , a_h, b_h\}$ for $S'$ with $\phi (a_\ell ) = i_\ell$ and $\phi (b_\ell ) = j_\ell$ for $1 \le \ell \le h$ if and only if the parity of the spin structure defined by these conditions agrees with the parity of the restriction to $S'$.

• (3) For any$(2h-2)$-tuple $(i_1, j_1, \dots ,i_{h-1}, j_{h-1})$ of elements of ${\mathbb {Z}}/ r {\mathbb {Z}}$, there is some geometric symplectic basis ${\mathcal {B}} = \{a_1, b_1, \dots , a_h, b_h\}$ for $S'$ with $\phi (a_\ell ) = i_\ell$ and $\phi (b_\ell ) = j_\ell$ for $1 \le \ell \le h-1$.

• (4) For a given 2h-tuple $(k_1, \dots , k_{2h})$ of elements of ${\mathbb {Z}}/r{\mathbb {Z}}$, there is some chain $(a_1, \dots , a_{2h})$ of curves on $S'$ such that $\phi (a_\ell ) = k_\ell$ for all $1 \le \ell \le 2h$ if and only if the parity of the spin structure defined by these conditions agrees with the parity of the restriction to $S'$.

• (5) For any$(2h-2)$-tuple $(k_1, \dots , k_{2h-2})$ of elements of ${\mathbb {Z}}/r{\mathbb {Z}}$, there is some chain $(a_1, \dots , a_{2h-2})$ of curves on $S'$ such that $\phi (a_\ell ) = k_\ell$ for all $1 \le \ell \le 2h-2$.

Proof

The proof is essentially identical to that of Corollary 4.5. The arguments for (2) and (3) are slightly novel; the remaining points follow their counterparts in Corollary 4.5 verbatim. To establish (2), let ${\mathcal {B}}' = \{a_\ell ', b_\ell '\}$ be a geometric symplectic basis on $S'$. Let $S''$ be a subsurface of $S'$ containing each curve in ${\mathcal {B}}'$ that has only one boundary component. Given $(i_1, j_1, \dots , i_h, j_h)$, there is some spin structure $\psi$ on $S''$ for which $\psi (a'_\ell ) = i_\ell$ and $\psi (b'_\ell ) = j_\ell$ for $1 \le \ell \le h$. By Proposition 4.9, there is an element $f \in {{\,\mathrm{Mod}\,}}(S'')$ for which $f^*(\psi ) = \phi$ if and only if the Arf invariants of $\phi$ and $\psi$ agree; if they do, then ${\mathcal {B}} = f^{-1}({\mathcal {B}}')$ has the required properties.

(3) will be obtained from (2). Let $\epsilon \in {\mathbb {Z}}/2{\mathbb {Z}}$ denote the Arf invariant of $\phi$, and define the quantity

$$\begin{aligned} \eta = \sum _{\ell = 1}^{h-1} (i_\ell + 1)(j_\ell +1) \pmod 2. \end{aligned}$$

As the formula (7) for the Arf invariant shows, given any $(2h-2)$-tuple $(i_1, j_i, \dots ,i_{h-1}, j_{h-1})$ and any value $\epsilon \in {\mathbb {Z}}/2{\mathbb {Z}}$, there is a choice of $i_h, j_h \in {\mathbb {Z}}/r{\mathbb {Z}}$ for which $\eta + (i_h +1)(j_h +1) \equiv \epsilon \pmod 2$. The result now follows by applying (2) to the tuple $(i_1, j_1, \dots , i_h, j_h)$. $\square$

We will also require a result establishing the existence of configurations ${\mathscr {D}}_n$ as in the $D_n$ relation (Proposition 2.6).

Corollary 4.11

Let $r = 2d$ be an even integer, and let $\Sigma _g$ be a closed surface endowed with a ${\mathbb {Z}}/2d{\mathbb {Z}}$-valued spin structure $\phi$. Let $\Delta$ be a curve on $\Sigma _g$ that separates $\Sigma _g$ into subsurfaces $S_1, S_2$ for which the genus $g(S_1) \ge d+1$. Set $n = 2g(S_1) - 2d + 1$. Then there exists a configuration $a, a', c_1, \dots , c_{n-2}$ of curves on $S_1$ arranged in the ${\mathscr {D}}_n$ configuration, such that the elements $a, a',$ and $c_i$ are admissible for all i, and such that $\Delta = \Delta _2$ as in Fig. 2.

Proof

By Corollary 4.10.5, there exists a chain $a, c_1, \dots , c_{n-2}$ of admissible curves on $S_1$. Let $a' \subset S_1$ be chosen so that $a \cup a' \cup \Delta$ bounds a subsurface of genus $g(S_1) - d - 1$ containing $c_i$ for $i \ge 2$, and such that $i(a',c_1) = 1$. The other side of $a \cup a'$ bounds a subsurface of genus d, and so the homological coherence property implies that $a'$ is admissible. By construction, the curves $a, a', c_1, \dots , c_{n-2}$ form the configuration ${\mathscr {D}}_{n}$ of the $D_n$ relation, and the boundary component $\Delta _2$ of Fig. 2 is given here by $\Delta$. $\square$

5 r odd: generating ${{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ]$ by Dehn twists

Let $\phi$ be a ${\mathbb {Z}}/r {\mathbb {Z}}$-valued spin structure on a closed surface $\Sigma _g$. Throughout this section we assume that $r \mid (2g-2)$ (so that, following Remark 3.6, $\Sigma _g$ admits a ${\mathbb {Z}}/r {\mathbb {Z}}$-valued spin structure) and that r is odd. Recall from Definition 3.16 that the admissible subgroup is defined via

$$\begin{aligned} {\mathcal {T}}_\phi = \langle T_a \mid a \text{ nonseparating } \text{ curve, } \phi (a) = 0 \rangle . \end{aligned}$$

By construction, ${\mathcal {T}}_\phi \leqslant {{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ]$. The main result of this section is that for r odd, this containment is an equality.

Proposition 5.1

For any $g \ge 3$ and for any odd integer r satisfying $r < g - 1$, there is an equality

$$\begin{aligned} {\mathcal {T}}_\phi = {{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ]. \end{aligned}$$

Before beginning with the proof, we will first establish some properties of the group ${\mathcal {T}}_\phi$ which will be used throughout this section and the next.

Lemma 5.2

Let $\phi$ be a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure on a surface $\Sigma _g$ with $r < g-1$ and $g \ge 5$. Let c be any nonseparating simple closed curve on $\Sigma _g$. Suppose that r is odd, or else that r is even and $\phi (c) \equiv 1 \pmod 2$. Then $T_c^r \in {\mathcal {T}}_\phi$.

Proof

Let c be as in the statement of Lemma 5.2. Our first objective is to construct a configuration of admissible curves ${\mathscr {D}}_{2r+3}$ as in Corollary 2.7 for which $c = C_k$. By hypothesis, there is an expression of the form $\phi (c) = 2k-1 \pmod r$ for some integer $1 \le k \le r$. Invoking Corollary 4.5.3 or 4.10.5 as appropriate, the hypothesis $r< g-1$ implies that there is a chain $a, c_1, \dots , c_{2k-1}$ of admissible curves disjoint from c, and there is a chain $c_{2k+1}, \dots , c_{2r+1}$ of admissible curves disjoint from c and from $a, c_1, \dots , c_{2k-1}$. Let $a'$ be a curve such that $a \cup a' \cup c$ bounds a surface of genus $k-1$ containing $c_2, \dots , c_{2k-1}$, and satisfying $i(a', c_1) = 1$ and $i(a', c_i) = 0$ for $2k+1 \le i \le 2r+1$. The homological coherence property implies that $a'$ is admissible.

To complete the construction, it remains only to find the curve $c_{2k}$. Such a curve $c_{2k}$ must be admissible, and $c_{2k}$ must have the following intersection properties:

$$\begin{aligned}&i(c_{2k}, c_{2k \pm 1}) = 1, \quad i(c_{2k}, a) = i(c_{2k}, a') = i(c_{2k}, c_i) = 0 \text{ for }\,\left| i-2k \right| > 1, \nonumber \\&\quad \quad i(c_{2k}, c) = 2. \end{aligned}$$

(8)

Let $c_{2k}'$ be any curve satisfying the intersection properties (8). If we can show that the complement of a regular neighborhood of the configuration ${\mathscr {D}}'_{2r+3} \,{:}{=}\, a, a', c_1, \dots , c_{2k-1}, c_{2k'}, c_{2k+1}, \dots , c_{2r+1}$ is a surface of positive genus, then the existence of $c_{2k}$ will follow from Corollary 4.4.

The configuration ${\mathscr {D}}'_{2r+3}$ is contained in a surface of genus $r+1$ with two boundary components. Each boundary component is homologous to the nonseparating curve c, so the complement has genus $g - r - 2$. We must show that this quantity is positive. Establishing $g -r - 2 \ge 1$ is a matter of simple arithmetic. Writing $r = \frac{2g-2}{m}$ for some $m \ge 3$, we have

$$\begin{aligned} g -r - 2= \frac{m-2}{m}(g-1) -1 \ge \frac{g}{3}-1>0, \end{aligned}$$

since $g \ge 5$ by hypothesis.

Recalling that the group $H_{2r + 3}^+$ from Corollary 2.7 is defined to be the group generated by the Dehn twists about the elements of ${\mathscr {D}}_{2 r + 3} \cup \{\Delta _1\}$, it follows that if each element of ${\mathscr {D}}_{2 r +3}$ is admissible, then $H_{2r+3}^+ \leqslant {\mathcal {T}}_\phi$. We have constructed the curves $a, a', c_1, \dots , c_{2r+1}$ so as to be admissible; homological coherence implies that also $\Delta _1$ is admissible. Corollary 2.7.2 then implies that $T_{C_k}^r \in {\mathcal {T}}_\phi$ for any $1 \le k \le r+1$. $\square$

Lemma 5.3

Let $\phi$ be a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure on a surface $\Sigma _g$, and let $v \in H_1(\Sigma _g; {\mathbb {Z}})$ be any primitive homology class. If r is odd, then for any $k \in {\mathbb {Z}}/r {\mathbb {Z}}$, there exists a curve c for which $[c] = v$ and $\phi (c) = k$. If r is even, then for any $k \in {\mathbb {Z}}/r {\mathbb {Z}}$ such that $\phi \pmod 2(v) \equiv k \pmod 2$, there exists a curve c for which $[c] = v$ and $\phi (c) = k$.

Proof

Let $c_0$ be any (oriented) curve on $\Sigma _g$ with $[c_0] = v$; set $\phi (c_0) = k_0$. Let $c_1$ be a curve disjoint from $c_0$ such that $c_0 \cup c_1$ bounds a subsurface of genus 1, oriented to the left of $c_0$. Then $\phi (c_1) = k_0+2$ when oriented with the subsurface to the right, and $[c_0] = [c_1]$. This construction can be repeated, giving rise to curves $c_m$ with $\phi (c_m) = k_0+2m$. If r is odd, then the set of values $k_0 + 2m$ for various values of m exhausts ${\mathbb {Z}}/r{\mathbb {Z}}$, and if r is even, then the set of values $k_0 + 2m$ exhausts the coset $k_0 + 2 {\mathbb {Z}}/r{\mathbb {Z}}$. The claim follows by taking $c = c_m$ for the appropriate value of m. $\square$

Proof of Proposition 5.1

The method is to compare the intersections of ${\mathcal {T}}_\phi$ and ${{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ]$ with ${\mathcal {I}}_g$ and ${\mathcal {K}}_g$. We first present a high-level overview of the logical structure of the proof that explains how Proposition 5.1 follows from ancillary results; these results are then obtained in Steps 1–4.

Overview Recall from (3) the symplectic representation $\Psi {:}\,{{\,\mathrm{Mod}\,}}(\Sigma _g) \rightarrow {{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})$ with kernel given by the Torelli group ${\mathcal {I}}_g$. To show that ${\mathcal {T}}_\phi = {{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ]$, it suffices to show that (I) $\Psi ({\mathcal {T}}_\phi ) = \Psi ({{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ])$ and that (II) ${\mathcal {T}}_\phi \cap {\mathcal {I}}_g = {{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ] \cap {\mathcal {I}}_g$.

The equality of (I) is obtained in Step 1 as Lemma 5.4. The proof of (II) is carried out in Steps 2–4. The method is to study the restriction of the Johnson homomorphism to the groups ${\mathcal {T}}_\phi \cap {\mathcal {I}}_g$ and ${{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ] \cap {\mathcal {I}}_g$. Recall from (4) that the Johnson homomorphism is the surjective homomorphism

$$\begin{aligned} \tau {:}\,{\mathcal {I}}_g \rightarrow \wedge ^3 H_{\mathbb {Z}}/H_{\mathbb {Z}}, \end{aligned}$$

and that the kernel is written ${\mathcal {K}}_g$. To establish (II), it suffices to show that (i) $\tau ({\mathcal {T}}_\phi \cap {\mathcal {I}}_g) = \tau ({{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ] \cap {\mathcal {I}}_g)$ and that (ii) ${\mathcal {T}}_\phi \cap {\mathcal {K}}_g = {{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ] \cap {\mathcal {K}}_g$. The equality of (i) is carried in Steps 2 and 3. The main result of Step 2, Lemma 5.7, establishes an upper bound on the image $\tau ({{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ] \cap \mathcal I_g)$, and the main result of Step 3, Lemma 5.8, shows that the subgroup $\tau ({\mathcal {T}}_\phi \cap {\mathcal {I}}_g)$ realizes this upper bound. Finally (ii) is established in Step 4: Lemma 5.9 shows that there is a containment ${\mathcal {K}}_g \leqslant {\mathcal {T}}_\phi$.

Step 1: The symplectic quotient The first step is to understand the image of ${\mathcal {T}}_\phi$ and ${{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ]$ in the symplectic group ${{\,\mathrm{Sp}\,}}(2g,{\mathbb {Z}})$.

Lemma 5.4

For r odd, the symplectic representation $\Psi {:}\,{{\,\mathrm{Mod}\,}}(\Sigma _g) \rightarrow {{\,\mathrm{Sp}\,}}(2g,{\mathbb {Z}})$ restricts to a surjection

$$\begin{aligned} \Psi {:}\,{\mathcal {T}}_{\phi } \twoheadrightarrow {{\,\mathrm{Sp}\,}}(2g,{\mathbb {Z}}). \end{aligned}$$

It follows that also $\Psi {:}\,{{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ] \twoheadrightarrow {{\,\mathrm{Sp}\,}}(2g,{\mathbb {Z}})$ is a surjection.

Proof

Let $v \in H_1(\Sigma _g; {\mathbb {Z}})$ be a primitive element. By Lemma 5.3, there is some curve c with $[c] = v$ and $\phi (c) = 0$. The result follows from this, since ${{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})$ is generated by the set of transvections $T_v$ given by $x \mapsto x + \langle x,v \rangle v$ for $v \in H_1(\Sigma _g; {\mathbb {Z}})$ primitive, and $\Psi (T_c) = T_{[c]}$. $\square$

Step 2:${{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ]$and the Johnson homomorphism Our next objective is Lemma 5.7 below. This concerns the image of ${{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ] \cap {\mathcal {I}}_g$ under the Johnson homomorphism. In order to formulate the result, it is necessary to first study a different quotient of ${\mathcal {I}}_g$ first constructed by Chillingworth [1, 2]. Chillingworth’s work is formulated using the notion of a “winding number function”; as explained in [9, Introduction], a winding number function is a particular instance of a spin structure. The properties of a winding number function that Chillingworth exploits in his work are common to all spin structures, and so we formulate his results in this larger context. See also [10, Section 5] for a brief summary of Chillingworth’s work. Recall in the statement below that ${\mathcal {S}}$ is defined to be the set of isotopy classes of oriented simple closed curves on a surface $\Sigma _g$.

Theorem 5.5

(Chillingworth) Let $\phi$ be a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure on a closed surface $\Sigma _g$. Let c be the function $c{:}\,{\mathcal {I}}_g \times \mathcal S \rightarrow {\mathbb {Z}}/r {\mathbb {Z}}$ defined by the formula

$$\begin{aligned} c(f, \gamma ) = \phi (f(\gamma )) - \phi (\gamma ). \end{aligned}$$

Then the value $c(f, \gamma )$ depends only on the homology class $[\gamma ] \in H_{\mathbb {Z}}$, and c descends to a homomorphism

$$\begin{aligned} c{:}\,{\mathcal {I}}_g \rightarrow {{\,\mathrm{Hom}\,}}(H_{\mathbb {Z}}, {\mathbb {Z}}/r {\mathbb {Z}}) \cong H^1(\Sigma _g; {\mathbb {Z}}/r {\mathbb {Z}}). \end{aligned}$$

In particular, c does not depend on the choice of ${\mathbb {Z}}/r {\mathbb {Z}}$-valued spin structure.

In [10], Johnson related Chillingworth’s homomorphism to the Johnson homomorphism. To formulate the precise connection, we require the following well-known lemma; see e.g [10, Sections 5,6].

Lemma 5.6

There is a ${{\,\mathrm{Sp}\,}}(2g,{\mathbb {Z}})$-equivariant surjection

$$\begin{aligned} C{:}\,\wedge ^3 H_{{\mathbb {Z}}} / H_{{\mathbb {Z}}} \rightarrow H_{{\mathbb {Z}}/(g-1){\mathbb {Z}}} \end{aligned}$$

given by the “contraction”

$$\begin{aligned} C(x\wedge y \wedge z) = \langle x,y \rangle z + \langle y,z \rangle x + \langle z,x \rangle y \pmod {g-1}. \end{aligned}$$

(9)

It follows that for any $r \mid (g-1)$, there is a ${{\,\mathrm{Sp}\,}}(2g,{\mathbb {Z}})$-equivariant surjection

$$\begin{aligned} C_r{:}\,\wedge ^3 H_{{\mathbb {Z}}} / H_{{\mathbb {Z}}} \rightarrow H_{{\mathbb {Z}}/r{\mathbb {Z}}} \end{aligned}$$

given by post-composing C with the reduction mod r. We can now formulate the main result of Step 3.

Lemma 5.7

Let $\phi$ be a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure on a surface of genus g, with $g \ge 3$ and r odd. Then $C_r \circ \tau = 0$ on ${{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ] \cap {\mathcal {I}}_g$.

Proof

According to [10, Theorem 3], the composition $C_r \circ \tau$ coincides (up to an application of Poincaré duality) with the mod-r Chillingworth homomorphism $c{:}\,{\mathcal {I}}_g \rightarrow H^1(\Sigma _g;{{\mathbb {Z}}/r {\mathbb {Z}}})$. The formula for c given in Theorem 5.5 shows that c measures how $f \in {\mathcal {I}}_g$ alters the set of values $\{\phi (\gamma )\mid \gamma \in {\mathcal {S}}\}$; it therefore follows immediately that the restriction of c to ${{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ] \cap {\mathcal {I}}_g$ is trivial. $\square$

Step 3${\mathcal {T}}_\phi$and the Johnson homomorphism In the previous step, we showed that there is a containment

$$\begin{aligned} \tau ({{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ] \cap {\mathcal {I}}_g) \leqslant \ker (C_r \circ \tau ). \end{aligned}$$

Our next result establishes that this containment is an equality, even when restricted to the subgroup $\tau ({\mathcal {T}}_\phi \cap {\mathcal {I}}_g)$.

Lemma 5.8

For $r< g-1$ odd and for $g \ge 3$, the Johnson homomorphism $\tau$ gives a surjection

$$\begin{aligned} \tau {:}\,{\mathcal {T}}_\phi \cap {\mathcal {I}}_g \twoheadrightarrow \ker (C_r \circ \tau ). \end{aligned}$$

Proof

Define $K \,{:}{=}\, \ker (C_r)$. We must show that ${\mathcal {T}}_\phi \cap {\mathcal {I}}_g$ surjects onto K under $\tau$. The first step will be to determine a generating set for K, and then we will exhibit each generator within $\tau ({\mathcal {T}}_\phi \cap {\mathcal {I}}_g)$.

To determine a generating set for K, we consider the short exact sequence

$$\begin{aligned} 1 \rightarrow K \rightarrow \wedge ^3 H_{\mathbb {Z}}/ H_{\mathbb {Z}}\rightarrow H_{{\mathbb {Z}}/r {\mathbb {Z}}} \rightarrow 1. \end{aligned}$$

determined by $C_r$. By lifting a set of relations $\{r_i\}$ for $H_{{\mathbb {Z}}/r{\mathbb {Z}}}$ to $\wedge ^3 H_{\mathbb {Z}}/H_{\mathbb {Z}}$, we will obtain a set of generators $\{{\tilde{r}}_i\}$ for K. Let ${\mathcal {B}} = \{x_1, y_1, \dots , x_g, y_g\}$ be a symplectic basis for $H_{\mathbb {Z}}$. There is an associated basis $\wedge ^3 {\mathcal {B}} \subset \wedge ^3 H_{\mathbb {Z}}$ given by

$$\begin{aligned} \wedge ^3 {\mathcal {B}} \,{:}{=}\, \left\{ z_1 \wedge z_2 \wedge z_3 \mid z_i \in {\mathcal {B}} \text{ distinct }\right\} \end{aligned}$$

Thus also $\wedge ^3H_{\mathbb {Z}}/H_{\mathbb {Z}}$ is generated by the image of $\wedge ^3 {\mathcal {B}}$.

To determine the relations $r_i$, we must understand $C_r(z_1 \wedge z_2 \wedge z_3)$ for the various possibilities for $\{z_1, z_2, z_3\}$. There are two orbits of generators under the action of ${{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})$. The first orbit consists of elements of the form $z \wedge x_i \wedge y_i$ (necessarily with $z \ne x_i, y_i$), and the second orbit consists of elements of the form $z_i \wedge z_j \wedge z_k$ with each $z_\ell \in \{x_\ell , y_\ell \}$ and with i, j, k mutually distinct.

The image of $z \wedge x_i \wedge y_i$ in $H_{{\mathbb {Z}}/r {\mathbb {Z}}}$ is

$$\begin{aligned} C_r(z \wedge x_i \wedge y_i) = z, \end{aligned}$$

while $C_r(z_i \wedge z_j \wedge z_k) = 0$ for elements of the second type. Define A to be the abelian group generated by the symbols $C_r(z_1 \wedge z_2 \wedge z_3)$ for $z_1 \wedge z_2 \wedge z_3 \in \wedge ^3 {\mathcal {B}}$, subject to the relations (R1)–(R3) below:

• (R1) $r C_r(z \wedge x_i \wedge y_i) = 0$

• (R2) $C_r(z \wedge x_i \wedge y_i) - C_r(z \wedge x_j \wedge y_j) = 0$

• (R3) $C_r(z_i \wedge z_j \wedge z_k) = 0$ for $\{i,j,k\} \subset \{1, \dots , g\}$ distinct.

It can be easily verified that there is an isomorphism $A \cong H_{{\mathbb {Z}}/r {\mathbb {Z}}}$, so that the relations (R1)–(R3) can be lifted to $\wedge ^3 H_{\mathbb {Z}}/ H_{\mathbb {Z}}$ to give a generating set for K as desired. The corresponding generators are given below.

• (G1) $r z \wedge x_i \wedge y_i$

• (G2) $z \wedge (x_i \wedge y_i - x_j\wedge y_j)$

• (G3) $z_i \wedge z_j \wedge z_k$ for $\{i,j,k\} \subset \{1, \dots , g\}$ distinct.

Having determined a generating set for K, it remains to exhibit each such generator in the form $\tau (f)$ for $f \in {\mathcal {T}}_\phi \cap {\mathcal {I}}_g$. These will be handled on a case-by-case basis. We start with (G1). By Lemma 2.8, there exist curves c, d that determine a genus-1 bounding pair map with $\tau (T_c T_d^{-1}) = z \wedge x_i \wedge y_i$. By Lemma 5.2, $T_c^r, T_d^r \in {\mathcal {T}}_\phi$, so that $T_c^r T_d^{-r} \in {\mathcal {T}}_\phi$ is an element with the required properties.

Fig. 5

The configuration of curves used to exhibit (G2)

Next we consider (G2). Let c be a curve with $[c] = z$ and $\phi (c) = 0$. By the change-of-coordinates principle, there exist curves a, b with the following properties: (1) $a \cup b$ bounds a subsurface S of genus 2, (2) $c \subset S$, (3) $[a] = [b] = [c]$, and c separates S into two subsurfaces $S_1, S_2$ each of genus 1, (4) $x_i, y_i$ determine a symplectic basis for $S_1$ and $x_j, y_j$ determine a symplectic basis for $S_2$. Such a configuration is shown in Fig. 5. By homological coherence, $\phi (a) = \phi (b) = -2$ when a, b are oriented with S to the left. By Lemma 2.8,

$$\begin{aligned} \tau (T_a T_c^{-1}) = z \wedge x_i \wedge y_i \end{aligned}$$

and

$$\begin{aligned} \tau (T_b T_c^{-1}) = -z \wedge x_j \wedge y_j. \end{aligned}$$

Therefore, it is necessary to show $T_a T_b T_c^{-2} \in \mathcal T_\phi$. By hypothesis, $T_c \in {\mathcal {T}}_\phi$. By Corollary 4.5.3, there exists a chain $a_1, \dots , a_5$ of curves on S for which $\phi (a_i) = 0$. By the chain relation (Proposition 2.4), $T_a T_b \in {\mathcal {T}}_\phi$, and the result follows.

Fig. 6

Top: The relevant portion of the geometric symplectic basis ${\mathcal {B}}$. Bottom: The configuration of curves used to exhibit (G3). Orientations have been suppressed wherever possible

It remains to exhibit generators of type (G3). Any such generator is equivalent under the action of ${{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})$ to $y_1 \wedge y_2 \wedge y_3$. By combining the ${{\,\mathrm{Sp}\,}}(2g,{\mathbb {Z}})$-equivariance of $\tau$ (Lemma 2.8.1) with the result of Lemma 5.4, it suffices to exhibit only $y_1 \wedge y_2 \wedge y_3$. Figure 6 shows the two 3-chains $C_1 = (c_61,c_2,c_3)$ and $C_2 = (c_1',c_2,c_1'+_\epsilon d)$. Observe that d is a boundary component for regular neighborhoods of both $C_1$ and $C_2$; let $e_1, e_2$ denote the other boundary component of $C_1, C_2$, respectively.

By Corollary 4.5.2, there exists a geometric symplectic basis ${\mathcal {B}}$ that contains the elements $c_1, c_2, b, f$ as depicted in the top portion of Fig. 6, with homology classes and $\phi$-values given in the table below. The remaining entries in the table have been filled in using the homological coherence property. (A value of $*$ indicates that the value is irrelevant and/or underdetermined, and if an orientation is left unspecified, this is in accordance with Convention 3.4).

By Lemma 2.8,

$$\begin{aligned}&\tau (T_d T_{e_1}^{-1})= x_1 \wedge y_1 \wedge y_2,\\&\tau (T_d T_{e_2}^{-1}) = (x_1 - y_3) \wedge y_1 \wedge y_2. \end{aligned}$$

It follows that $\tau (T_{e_1}^{-1}T_{e_2}) = y_3 \wedge y_1 \wedge y_2$. As $d \cup e_1$ and $d \cup e_2$ each bound subsurfaces of genus 1 and $\phi (d) = -2$ when d is oriented with these subsurfaces to the left, the homological coherence property implies that $e_1$ and $e_2$ are admissible. The result follows. $\square$

Step 4: The Johnson kernel The final piece of the analysis concerns the relationship between ${{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ]$ and the Johnson kernel ${\mathcal {K}}_g$.

Lemma 5.9

Let $\phi$ be a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure with r odd. If $g \ge 3$, then ${\mathcal {T}}_\phi$ contains the Johnson kernel $\mathcal K_g$. It follows that also

$$\begin{aligned} {\mathcal {K}}_g \leqslant {{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ]. \end{aligned}$$

Proof

According to Johnson’s Theorem 2.9, $\mathcal K_g$ has a generating set consisting of the set of all $T_c$ for c a separating curve. Each such c divides $\Sigma _g$ into two subsurfaces $S, S'$, and since $g \ge 3$, without loss of generality we can assume that $g(S) > 1$. By Corollary 4.5.3, there exists a chain $a_1, \dots , a_{2g(S)}$ of curves on S such that $\phi (a_i) = 0$ for all i. By hypothesis, $T_{a_i} \in {\mathcal {T}}_\phi$ for all i. By the chain relation (Proposition 2.4), it follows that $T_c \in {\mathcal {T}}_\phi$ as required. $\square$

This concludes the proof of Proposition 5.1. $\square$

6 r even: ${\mathcal {T}}_\phi$ has finite index in ${{\,\mathrm{Mod}\,}}(\Sigma _g)$

We continue to assume that $r \mid (2g-2)$, but now we take $r = 2d$ to be even. For r even, we cannot give a complete characterization of ${\mathcal {T}}_\phi$ as in Proposition 5.1, but we will show that ${\mathcal {T}}_\phi$ has finite index in ${{\,\mathrm{Mod}\,}}(\Sigma _g)$. The minimal genus for which the ensuing arguments apply has a rather intricate dependence on r, encapsulated in the definition below.

Definition 6.1

For an integer $d \ge 1$, define k(d) as follows:

$$\begin{aligned} k(d) = {\left\{ \begin{array}{ll} 2 &{}\quad d \text{ odd } \text{ or } d \ge 6 \text{ even }\\ 6 &{}\quad d =2\\ 5 &{}\quad d = 4. \end{array}\right. } \end{aligned}$$

Suppose $r = 2d$ is an even integer. Define

$$\begin{aligned} g(r) = k(d) d + 1. \end{aligned}$$

Proposition 6.2

Let $r = 2d$ be an even integer. Suppose $g \ge g(r)$ and that $r < g - 1$. Then ${\mathcal {T}}_\phi$ is a finite-index subgroup of ${{\,\mathrm{Mod}\,}}(\Sigma _g)$.

The presence of an underlying ${\mathbb {Z}}/2{\mathbb {Z}}$ spin structure makes proving the analogues of Lemmas 5.8 and 5.9 substantially more difficult. At present, we do not know how to establish the analogue of Lemma 5.8, owing to the fact that the Arf invariant provides an obstruction to finding the configurations of curves on subsurfaces needed for the arguments therein. Thus we content ourselves with showing that ${\mathcal {T}}_\phi \leqslant {{\,\mathrm{Mod}\,}}(\Sigma _g)$ is finite-index.

Proof of of Proposition 6.2

The proof of Proposition 6.2 follows a similar outline to that of Proposition 5.1. We begin with an overview of the proof.

Overview To establish finiteness of the index $[{{\,\mathrm{Mod}\,}}(\Sigma _g){:}\,{\mathcal {T}}_\phi ]$, it suffices to show that the indices $[{{\,\mathrm{Sp}\,}}(2g,{\mathbb {Z}}){:}\,\Psi ({\mathcal {T}}_\phi )]$ and $[{\mathcal {I}}_g{:}\,{\mathcal {T}}_\phi \cap {\mathcal {I}}_g]$ are both finite. Finiteness of $[{{\,\mathrm{Sp}\,}}(2g,{\mathbb {Z}}){:}\,\Psi ({\mathcal {T}}_\phi )]$ is established in Lemma 6.4 of Step 1, which moreover gives a complete description of the subgroup $\Psi ({\mathcal {T}}_\phi )$.

Finiteness of $[{\mathcal {I}}_g{:}\,{\mathcal {T}}_\phi \cap {\mathcal {I}}_g]$ is obtained in Steps 2 and 3, again by using the Johnson homomorphism to analyze the intersection ${\mathcal {T}}_\phi \cap {\mathcal {I}}_g$ as in Steps 2-4 of the proof of Proposition 5.1. The main result of Step 2 is Lemma 6.6, which shows that $\tau ({\mathcal {T}}_\phi \cap {\mathcal {I}}_g)$ has finite index in $\wedge ^3 H_{\mathbb {Z}}/H_{\mathbb {Z}}$. Step 3 completes the argument by showing the containment ${\mathcal {K}}_g \leqslant {\mathcal {T}}_\phi$; this is obtained as Lemma 6.7. We advise the reader that Step 3 is substantially more complicated than its counterpart Step 4 of the proof of Proposition 5.1, and will require an explanatory outline of its own.

Step 1: The symplectic quotient The case of r even is no more difficult than for r odd. Let q be a ${\mathbb {Z}}/2{\mathbb {Z}}$-valued spin structure. An anisotropic transvection is a transvection

$$\begin{aligned} T_v(w) = w + \langle w,v \rangle v \end{aligned}$$

for a primitive $v \in H_1(\Sigma _g; {\mathbb {Z}})$ such that $q(v) = 0$.

The following theorem is surely well-known to experts but we were unable to find a reference. A special case is treated in [5, Proposition 14].

Theorem 6.3

(Folklore) Let q be a ${\mathbb {Z}}/2{\mathbb {Z}}$-valued spin structure on $\Sigma _g$ for $g \ge 3$, and let ${{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})[q]$ denote the subgroup of ${{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})$ that fixes q. Then ${{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})[q]$ is generated by the collection of anisotropic transvections

$$\begin{aligned} \{T_v \mid v \in H_1(\Sigma _g; {\mathbb {Z}}) \text{ primitive, } q(v) = 0\}. \end{aligned}$$

Proof

The action of ${{\,\mathrm{Sp}\,}}(2g,{\mathbb {Z}})$ on the set of spin structures factors through the quotient $f{:}\,{{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}}) \rightarrow {{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}}/2{\mathbb {Z}})$. Define ${{\,\mathrm{Sp}\,}}(2g,{\mathbb {Z}})[2] \,{:}{=}\, \ker (f)$. Thus, there is a short exact sequence

$$\begin{aligned} 1 \rightarrow {{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})[2] \rightarrow {{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})[q] \rightarrow {{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}}/2{\mathbb {Z}})[q] \rightarrow 1, \end{aligned}$$

with ${{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}}/2{\mathbb {Z}})[q]$ denoting the stabilizer of q in ${{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}}/2{\mathbb {Z}})$. According to [8, Theorem 14.16], the group ${{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}}/2{\mathbb {Z}})[q]$ is generated by the images of all anisotropic transvections. So it remains to see only that the subgroup of ${{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})[q]$ generated by anisotropic transvections contains ${{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})[2]$. According to [14, Lemma 5], the group ${{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})[2]$ is generated by the collection of “square transvections” $T_w^2$, where w ranges over all primitive $w \in H_1(\Sigma _g; {\mathbb {Z}})$.

If $q(w) = 0$ then $T_w \in {{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})[q]$ and so there is nothing to show. Assume now that $q(w) = 1$. It is easy to produce (e.g. by the change-of-coordinates principle on $\Sigma _g$) vectors $v_1, v_2, v_3 \in H_1(\Sigma _g; {\mathbb {Z}})$ with the following properties:

• (1) $q(v_i) = 0$ for all i,

• (2) $\langle v_1,v_2 \rangle = \langle v_2,v_3 \rangle = 1$ and $\langle v_1, v_3 \rangle = 0$,

• (3) $\langle v_i, w \rangle = 0$ for all i,

• (4) $v_1 + v_3 = w$.

The chain relation in ${{\,\mathrm{Mod}\,}}(\Sigma _g)$ (Proposition 2.4) descends to show the relation

$$\begin{aligned} (T_{v_1}T_{v_2}T_{v_3})^4 = T_w^2. \end{aligned}$$

Since the left-hand side is a product of anisotopic transvections, it follows that for w arbitrary, $T_w^2 \in {{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})[q]$ as required. $\square$

The following is the main result of Step 1.

Lemma 6.4

Let $\phi$ be a ${\mathbb {Z}}/r {\mathbb {Z}}$-valued spin structure for r an even integer, and let

$$\begin{aligned} q \,{:}{=}\, \phi \pmod 2 \end{aligned}$$

denote the associated ${\mathbb {Z}}/2{\mathbb {Z}}$-valued spin structure. The symplectic representation $\Psi {:}\,{{\,\mathrm{Mod}\,}}(\Sigma _g) \rightarrow {{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})$ restricts to a surjection

$$\begin{aligned} \Psi {:}\,{\mathcal {T}}_\phi \twoheadrightarrow {{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})[q], \end{aligned}$$

where ${{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})[q]$ denotes the stabilizer of q in ${{\,\mathrm{Sp}\,}}(2g,{\mathbb {Z}})$.

Proof

As ${\mathcal {T}}_\phi$ preserves the ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure $\phi$, it also preserves the mod-2 reduction q. Thus $\mathcal T_\phi \leqslant {{\,\mathrm{Mod}\,}}(\Sigma _g)[q]$ and so $\Psi ({\mathcal {T}}_\phi ) \leqslant {{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})[q]$. Let $v \in H_1(\Sigma _g; {\mathbb {Z}})$ be a primitive element satisfying $q(v) = 0$. By Lemma 5.3, there is some curve c with $[c] = v$ and $\phi (c) = 0$. As $T_c \in {{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ]$ and $\Psi (T_c) = T_v$, the result now follows from Theorem 6.3. $\square$

Step 2: The Johnson homomorphism We remind the reader that while the value $\phi (c)$ on a simple closed curve depends on more than the homology class $[c] \in H_1(\Sigma _g;{\mathbb {Z}})$, the discussion of Sect. 3.4 establishes that the mod-2 reduction q(c) does depend only on the homology class [c] (indeed, the coefficients here can be taken to be ${\mathbb {Z}}/2{\mathbb {Z}}$). Thus the arguments in Step 3 can be carried out entirely in the homological setting.

For the duration of Step 2, we adopt the following notation. As usual, define

$$\begin{aligned} q\,{:}{=}\, \phi \pmod 2. \end{aligned}$$

There exists a symplectic basis $\{x_1, y_1, \dots , x_g, y_g\}$ for $H_1(\Sigma _g;{\mathbb {Z}})$ such that $q(x_i) = 0$ for $1 \le i \le g$ and $q(y_j) = 0$ for $1 \le j \le g-1$; with such a basis, ${{\,\mathrm{Arf}\,}}(q)$ depends only on g and on $q(y_g)$.

Before proceeding to the main result of Step 2 (Lemma 6.6), we begin with an algebraic lemma.

Lemma 6.5

Set $v\,{:}{=}\, x_1 \wedge y_1 \wedge x_4$. Let $V \leqslant \wedge ^3 H_{\mathbb {Z}}$ denote the submodule generated by the set

$$\begin{aligned} \left\{ gv \mid g \in {{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})[q]\right\} . \end{aligned}$$

Then $V = \wedge ^3 H_{\mathbb {Z}}$ for $g \ge 5$.

Proof

As remarked in Lemma 5.8, $\wedge ^3H_{\mathbb {Z}}$ is generated by elements of the form $z_i \wedge z_j \wedge z_k$ with each $z_i \in \{x_1, y_1,\dots , x_g, y_g\}$. To begin with, we will exhibit generators for the submodule of $\wedge ^3 H_{\mathbb {Z}}$ spanned by generators $z_i\wedge z_j \wedge z_k$ for which $z_i,z_j,z_k \in \{x_1,y_1, \dots ,x_{g-1}, y_{g-1}\}$. The restriction of ${{\,\mathrm{Sp}\,}}(2g;{\mathbb {Z}})[q]$ to this submodule is independent of the parity of q. For $i \ne j \le g-1$, define $S_{i,j} \in {{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})$ via

$$\begin{aligned} S_{i,j}(x_i)&= x_j,\quad S_{i,j}(y_i)= y_j,\\ S_{i,j}(x_j)&= x_i,\quad S_{i,j}(y_j) = y_i, \end{aligned}$$

with all other generators fixed. As $q(x_k) = q(y_k) = 0$ for $k \le g-1$, in fact $S_{i,j}$ is an element of ${{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})[q]$. Applying $S_{i,j}$ for $i,j \ne 4$ to v shows that V contains all generators of the form $x_i \wedge y_i \wedge x_4$. Applying $S_{i,4}$ to $x_j \wedge y_j \wedge x_4$ for $i \ne j$ shows that V contains all generators of the form $x_j \wedge y_j \wedge x_i$ for $j \ne 4$; then applying $S_{j,4}$ to $x_j \wedge y_j \wedge x_i$ shows that V contains all elements of the form $x_j \wedge y_j \wedge x_i$.

For $1 \le i \le g-1$, define $R_i \in {{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})$ via

$$\begin{aligned} R_i(x_i) = y_i,\quad R_i(y_i)= -x_i \end{aligned}$$

with all other generators fixed. Again, the condition $q(x_k) = q(y_k) = 0$ for $k \le g-1$ implies that $R_i$ is an element of ${{\,\mathrm{Sp}\,}}(2g,{\mathbb {Z}})[q]$. Applying $R_i$ to $x_j \wedge y_j \wedge x_i$ shows that also V contains all elements of the form $x_j \wedge y_j \wedge y_i$.

It remains to exhibit generators of the form $z_i \wedge z_j \wedge z_k$ with $z_\ell \in \{x_\ell ,y_\ell \}$ and $i,j,k<g$ all distinct. Consider the transvection $T_{x_4 - x_1} \in {{\,\mathrm{Sp}\,}}(2g,{\mathbb {Z}})[q]$. Applied to $x_1 \wedge y_1 \wedge x_2$, this shows that

$$\begin{aligned} x_1 \wedge (y_1 + x_4) \wedge x_2 \in V, \end{aligned}$$

hence also $x_1 \wedge x_2 \wedge x_4 \in V$. Now by repeated applications of the elements $S_{i,j}$ and $R_i$, one can produce all remaining generators.

In the case $q(y_g) = 0$, the elements $S_{i,g}$ and $R_g$ are contained in ${{\,\mathrm{Sp}\,}}(2g,{\mathbb {Z}})[q]$, and so the above argument extends to complete this case. It remains to consider the case where $q(y_g) = 1$. In this case, the formula (5) defining a ${\mathbb {Z}}/2{\mathbb {Z}}$-valued quadratic form shows that $q(y_{g-1} + y_{g}) = 0$. Applying $T_{y_{g-1} + y_{g}}$ to the elements $x_1 \wedge x_2 \wedge x_{g-1}$ and $x_1 \wedge y_1 \wedge x_{g-1}$ shows that $x_1 \wedge x_2 \wedge y_g$ and $x_1 \wedge y_1 \wedge x_g$ are elements of V. Applying $S_{i,j}$ and $R_i$ for $i,j \le g-1$ produces all elements of the form $z_i \wedge z_j \wedge y_g$ with $z_\ell \in \{x_\ell ,y_\ell \}\ (i,j \le g-1)$. Then applying $T_{x_g}$ to these elements shows that also each $z_i \wedge z_j \wedge x_g \in V$.

By (5), we have $q(x_1 + x_g - y_g) = 0$. Applying $T_{x_1 + x_g - y_g}^{-1}$ to $y_1 \wedge y_2 \wedge y_g$ gives

$$\begin{aligned} w = (y_1 + x_1 + x_g - y_g) \wedge y_2 \wedge (x_1 + x_g); \end{aligned}$$

expanding this product yields the expression $w = -y_2 \wedge x_g \wedge y_g + w'$, with $w'$ expressed entirely in terms of generators already known to be elements of V. Applying $S_{i,j}$ and $R_{i}$ as in the above paragraph shows that all the remaining generators $z_i \wedge x_g \wedge y_g$ are elements of V. $\square$

The following is the main result of Step 2.

Lemma 6.6

For $g \ge 5$, the image $\tau ({\mathcal {T}}_\phi \cap {\mathcal {I}}_g)$ under the Johnson homomorphism is a finite-index subgroup of $\wedge ^3 H_{\mathbb {Z}}/H_{\mathbb {Z}}$.

Proof

As stated in Lemma 2.8.1, the homomorphism $\tau {:}\,{\mathcal {I}}_g \rightarrow \wedge ^3 H_{\mathbb {Z}}/H_{\mathbb {Z}}$ is ${{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})$-equivariant. The strategy for the proof of Lemma 6.6 is to first exhibit a single nonzero element of $\tau ({\mathcal {T}}_\phi \cap {\mathcal {I}}_g)$, and then to exploit this equivariance.

By Corollary 4.10.5, there exists a 3-chain of admissible curves $a_1, a_2, a_3$ such that

$$\begin{aligned} {[}a_1] = x_1,\quad [a_2] = y_1, \quad [a_3] = x_4- x_1. \end{aligned}$$

Let $\nu$ be a regular neighborhood of this chain, and denote the boundary curves as $b, b'$. As $a_1$ and $a_2$ are admissible, homological coherence implies that $\phi (b) = \phi (b') = -1$ when oriented so that $\nu$ lies to the left of both $b, b'$. By Lemma 5.2, $T_b^r$ is an element of ${\mathcal {T}}_\phi$. It follows by the chain relation (Proposition 2.4) that the bounding pair map $T_b^r T_{b'}^{-r} \in {\mathcal {T}}_\phi$. One sees that $[b] = [a_1]+ [a_3] = x_4$. By Lemma 2.8.3,

$$\begin{aligned} \tau (T_b^r T_{b'}^{-r}) = r (x_1 \wedge y_1 \wedge x_4). \end{aligned}$$

By Lemma 6.4 and the equivariance of $\tau$ with respect to ${{\,\mathrm{Sp}\,}}(2g,{\mathbb {Z}})$ (and a fortiori with respect to ${{\,\mathrm{Sp}\,}}(2g,{\mathbb {Z}})[q]$), it follows that $\tau ({\mathcal {T}}_\phi \cap {\mathcal {I}}_g)$ contains the ${\mathbb {Z}}$-span of the entire ${{\,\mathrm{Sp}\,}}(2g, {\mathbb {Z}})[q]$-orbit of $v\,{:}{=}\, r (x_1 \wedge y_1 \wedge x_4)$. Lemma 6.6 now follows from Lemma 6.5. $\square$

Step 3: The Johnson kernel In this section, we establish the following result.

Lemma 6.7

Let $\phi$ be a ${\mathbb {Z}}/2d{\mathbb {Z}}$-valued spin structure on $\Sigma _g$. Assume that g satisfies the hypotheses of Proposition 6.2. Then ${\mathcal {T}}_\phi$ contains the Johnson kernel ${\mathcal {K}}_g$.

Before beginning the proof, we explain the difficulties imposed by the assumption that $r = 2d$ is even.

The Arf invariant as obstruction The mechanism of proof for Lemma 5.9 was the chain relation (Proposition 2.4): if $S\subset \Sigma _g$ has one boundary component, we exploited Corollary 4.5 to produce a maximal chain $\{a_i\}$ of curves on S with $\phi (a_i) = 0$, and then used the chain relation to express $T_{\partial S}$ in terms of the admissible twists $\{T_{a_i}\}$. Now suppose $\phi$ is a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure for r even, and let $q = \phi \pmod 2$ denote the mod-2 reduction. For any subsurface $S\subset \Sigma _g$ with one boundary component, q restricts to give a ${\mathbb {Z}}/2{\mathbb {Z}}$-valued spin structure on S. The Arf invariant of , written here as $\epsilon (S)$, provides an obstruction to the existence of a maximal chain $\{a_i\}$ of admissible curves on S, since such a chain determines the value $\epsilon (S)$ solely as a function of g(S).

Suppose $c \subset \Sigma _g$ is a separating curve that divides $\Sigma _g$ into disjoint surfaces $S, S'$. Such a c is called easy if at least one of $S, S'$ supports a maximal chain of admissible curves, and is hard otherwise. By Corollary 4.10.4 and the chain relation (Proposition 2.4), if c is easy, then $T_c \in {\mathcal {T}}_\phi$.

Outline of proof of Lemma 6.7 We begin with Lemma 6.8, which characterizes those subsurfaces supporting a maximal chain of admissible curves in terms of the Arf invariant. This in particular shows the relevance of the genus of the subsurface mod 4, which in turn forces us to treat the cases $r \equiv 0, r \equiv 2 \pmod 4$ separately. We therefore establish Lemma 6.7 by combining Lemmas 6.10 and 6.13, which treat the cases of $r \equiv 2 \pmod 4$ and $r \equiv 0 \pmod 4$, respectively.

These are handled in Substeps 1 and 2, respectively. In each case, we first show that all separating twists of particular genera are elements of ${\mathcal {T}}_\phi$. In Substep 1, Lemma 6.9 shows that all separating twists of genus d lie in $\mathcal T_\phi$. In Substep 2, Lemma 6.11 shows that all separating twists of genus $h \equiv d+2 \pmod {2d}$ lie in $\mathcal T_\phi$, and Lemma 6.12 establishes the same result for separating twists of genus $h \equiv d+4 \pmod {2d}$. Lemmas 6.10 and 6.13 then follow from these preliminary results and an application of the $D_n$ relation (Proposition 2.6).

Lemma 6.8

Let $S \subset \Sigma$ be a subsurface with single boundary component. Assume the genus $g(S) \ge 2$. Then there is a maximal chain of admissible curves on S if and only if one of the following conditions hold:

• $g(S) \equiv 1 \text{ or } 2 \pmod 4$ and $\epsilon (S) = 1$,

• $g(S) \equiv 3 \text{ or } 0 \pmod 4$ and $\epsilon (S) = 0$.

Proof

Suppose S supports a maximal chain $a_1, \dots , a_{2g(S)}$ of admissible curves. Since the chain determines a basis for $H_1(S; {\mathbb {Z}})$, the conditions $\phi (a_i) = 0$ completely determine $\phi$. One can easily compute $\epsilon (S)$ from this and see that the above conditions are necessary. Sufficiency follows from Corollary 4.10.4. $\square$

Substep 1: d odd The objective of Substep 1 is Lemma 6.10 below. The first step is to see that all separating twists $T_c$ of genus d are elements of ${\mathcal {T}}_\phi$, regardless of whether c is easy.

Lemma 6.9

Let $S \subset \Sigma _g$ be a subsurface of genus d with a single boundary component c. If $\phi$ is a ${\mathbb {Z}}/2d{\mathbb {Z}}$-valued spin structure with d odd, then $T_c \in {\mathcal {T}}_\phi$.

Proof

If c is easy then there is nothing to show. Assume therefore that c is hard. If c is oriented so that S lies to the right, then $\phi (c) = -(1-2d) \equiv -1 \pmod {2d}$. The assumption that $r = 2d < g-1$ implies that $\Sigma _g {\setminus } S$ has genus at least 2. We claim that there exists a 3-chain of admissible curves x, y, z on $\Sigma _g {\setminus } S$ such that $c \cup x \cup z$ forms a pair of pants. To see this, we invoke Corollary 4.3 to let $x \subset \Sigma _g {\setminus } S$ be an admissible curve. Let $z \subset \Sigma _g {\setminus } S$ be any curve such that $c \cup x \cup z$ bounds a pair of pants; admissibility of z follows by the homological coherence property, as c is oriented with $\Sigma _g {\setminus } S$ to the left. To construct y, let $y' \subset \Sigma _g {\setminus } S$ be any curve such that $x, y', z$ forms a chain. By Corollary 4.4, $y'$ can be replaced with an admissible curve y with the same intersection properties.

Let $S'$ denote the connected surface of genus $d+1$ containing S and $x \cup y \cup z$. If ${\mathcal {B}}$ is a basis for $H_1(S; {\mathbb {Z}})$, then ${\mathcal {B}} \cup \{x,y\}$ forms a basis for $H_1(S';{\mathbb {Z}})$. Applying the formula (7) for the Arf invariant, it follows that $\epsilon (S') = \epsilon (S)+1$.

Since c is hard and $d=g(S)$ is odd, Lemma 6.8 implies that $\epsilon (S) = 0$ if $g(S) \equiv 1 \pmod 4$ and that $\epsilon (S) = 1$ otherwise. Recalling that $\epsilon (S') = \epsilon (S) + 1$, in the first case, $g(S') \equiv 2 \pmod 4$ and $\epsilon (S') = 1$, and in the second case, $g(S') \equiv 0 \pmod 4$ and $\epsilon (S') = 0$. Lemma 6.8 then implies that $c'\,{:}{=}\, \partial S'$ must be easy, and so $T_{c'} \in \mathcal T_\phi$. Applying the chain relation (Proposition 2.4) to x, y, z shows that $T_c T_{c'} \in {\mathcal {T}}_\phi$; this implies that also $T_c \in {\mathcal {T}}_\phi$. $\square$

Lemma 6.10

Let $\phi$ be a ${\mathbb {Z}}/2d{\mathbb {Z}}$-valued spin structure on $\Sigma _g$ with d odd. Assume that g satisfies the hypotheses of Proposition 6.2. Then ${\mathcal {T}}_\phi$ contains the Johnson kernel ${\mathcal {K}}_g$.

Proof

By Theorem 2.9, it suffices to show that $T_c \in {\mathcal {T}}_\phi$ for all separating curves c of arbitrary genus. To do this, we combine Lemma 6.9 with the $D_n$ relation (Proposition 2.6). Suppose c is a separating curve on $\Sigma _g$. Since $g = kd+1$ with $k \ge 2$, at least one side of c must be a subsurface S of genus $g(S) \ge d+1$. Set $n \,{:}{=}\, 2g(S) -2d +1$. By Corollary 4.11, there is a configuration ${\mathscr {D}}_n$ of admissible curves as in the $D_n$ relation for which $\Delta _2 = c$. The other boundary component $\Delta _0$ bounds a subsurface of genus d. Applying the $D_n$ relation, we have $T_{\Delta _0}^{n-1} T_c \in \mathcal T_\phi$. But since $\Delta _0$ bounds a surface of genus d, also $T_{\Delta _0} \in {\mathcal {T}}_\phi$ by Lemma 6.9. Thus ${\mathcal {K}}_g \leqslant {\mathcal {T}}_\phi$ in this case. $\square$

Substep 2: d even The objective is to establish Lemma 6.13. The argument here follows a similar outline to that of Substep 1 but now requires the two preliminary Lemmas 6.11 and 6.12.

Lemma 6.11

Let $S \subset \Sigma _g$ be a subsurface of genus $g(S) \ge 5$ with a single boundary component c, such that $g(S) \equiv d + 2 \pmod {2d}$. If $\phi$ is a ${\mathbb {Z}}/2d{\mathbb {Z}}$-valued spin structure with d even, then $T_c \in {\mathcal {T}}_\phi$.

Proof

Orient c so that S lies to the left. Then

$$\begin{aligned} \phi (c) = 1-2g(S) \equiv 1 - 2(d+2) \equiv -3 \pmod {2d}. \end{aligned}$$

By Corollary 4.10.5, there exists a chain $a_1, \dots , a_6$ of admissible curves on S. Let $a_7$ be any curve on S such that $i(a_7, a_k) = 1$ for $k = 6$ and is zero for $k \le 5$, and such that $c \cup a_1 \cup a_3 \cup a_5 \cup a_7$ bounds a subsurface of S homeomorphic to $\Sigma _{0,5}$. By homological coherence, $a_7$ is admissible.

Let $S'$ denote the subsurface of S homeomorphic to $\Sigma _{g(S)-3, 1}$ determined by the complement of the chain $a_1, \dots , a_7$. Applying the formula (7) for the Arf invariant, one finds that $\epsilon (S') = \epsilon (S)$. On the other hand, $g(S') \equiv g(S)+1 \pmod {4}$. By hypothesis, g(S) is even, and so referring to Lemma 6.8, if c is hard, then $c'\,{:}{=}\, \partial S'$ must be easy. The arguments given at the conclusion of Lemma 6.9 now apply to give the result. $\square$

Lemma 6.12

Let $S \subset \Sigma$ be a subsurface of genus $g(S) \ge 9$ with a single boundary component c, such that $g(S) \equiv d+4 \pmod {2d}$. If $\phi$ is a ${\mathbb {Z}}/2d{\mathbb {Z}}$-valued spin structure with d even, then $T_c \in {\mathcal {T}}_\phi$.

Proof

This is proved along similar lines to Lemma 6.11. Arguing as in the first paragraph of the proof of Lemma 6.11, there exists a chain $a_1, \dots , a_{15}$ of admissible curves on S such that $c \cup a_1 \cup a_3 \cup \dots \cup a_{15}$ bounds a subsurface of S homeomorphic to $\Sigma _{0,9}$. Let $S'$ denote the subsurface of S homeomorphic to $\Sigma _{g(S) - 7,1}$ determined by the complement of the chain $a_1, \dots , a_{15}$. The rest of the argument proceeds as in Lemma 6.11: one shows that if c is hard, necessarily $c' \,{:}{=}\, \partial S'$ must be easy, and the result follows as before by the chain relation (Proposition 2.4). $\square$

Lemma 6.13

Let $\phi$ be a ${\mathbb {Z}}/2d{\mathbb {Z}}$-valued spin structure on $\Sigma _g$ with d even. Assume that g satisfies the hypotheses of Proposition 6.2. Then ${\mathcal {T}}_\phi$ contains the Johnson kernel ${\mathcal {K}}_g$.

Proof

According to Johnson’s Theorem 2.9, in order to show that ${\mathcal {K}}_g \leqslant {\mathcal {T}}_\phi$, it suffices to exhibit all separating twists of genus 1 and 2 as elements of ${\mathcal {T}}_\phi$. To do this, we again appeal to the $D_n$ relation (Proposition 2.6). Suppose c is a separating curve on $\Sigma _g$ with $g(c) \le 2$. By hypothesis, $g \ge kd+1$ with d even and $k \ge 2$. Since the genus of one side of c is at most 2, the genus h of the other side of c is at least $kd - 1 \ge 2d-1$. If $d \ge 6$, then $2d-1 \ge 11$. If $d = 4$, then by assumption $k \ge 5$, and so $h \ge 19$. If $d = 2$ then we assume $k \ge 6$, so that $h \ge 11$.

In all three of these cases, Corollary 4.11 implies that there exists an $n \ge 4$ and a configuration $a, a', c_1, \dots , c_{2n-1}$ of admissible curves in the configuration of the $D_{2n+1}$ relation, with $\Delta _2 = c$ and $C_4$ bounding a subsurface of genus $g(S) \equiv d+4 \pmod {2d}$ disjoint from S, such that the hypotheses of Lemmas 6.11 and 6.12 hold.

By the $D_{k}$ relation (for $k = 2n+1,5,9$ respectively), $T_{\Delta _0}^{2n-1}T_c$ and $T_{\Delta _0}^3 T_{C_2}$ and $T_{\Delta _0}^7 T_{C_4}$ are all elements of ${\mathcal {T}}_\phi$. By Lemma 6.11, $T_{C_2} \in {\mathcal {T}}_\phi$ as well, hence $T_{\Delta _0}^3 \in {\mathcal {T}}_\phi$. Likewise, Lemma 6.12 shows that $T_{C_4} \in {\mathcal {T}}_\phi$, hence $T_{\Delta _0}^7 \in {\mathcal {T}}_\phi$. Combining these last two results shows that $T_{\Delta _0} \in {\mathcal {T}}_\phi$, and ultimately that $T_c \in {\mathcal {T}}_\phi$ as required. $\square$

This concludes the proof of Proposition 6.2. $\square$

7 Connectivity of some complexes

This section is devoted to establishing the connectivity of the simplicial complexes ${\mathcal {C}}_{sep,2}(\Sigma _g)$ and $\mathcal C_\phi ^1(\Sigma _g)$ to be defined below. The first of these will be an important ingredient in the proof of Proposition 8.2, and the second will feature in the proof of Theorem A. The mechanism by which these will be seen to be connected is the so-called Putman trick. The version given below is slightly less general than the full theorem as stated in [16], but will suffice for our purposes.

Theorem 7.1

(The Putman trick) Let X be a simplicial graph, and let G act on X by simplicial automorphisms. Suppose that the action of G on the set of vertices $X^{(0)}$ is transitive. Fix some base vertex $v \in X^{(0)}$. Let $\Sigma = \Sigma ^{-1}$ be a symmetric set of generators for G, and suppose that for each $s \in \Sigma$, there is a path in X connecting v to $s\cdot v$. Then X is connected.

Definition 7.2

${\mathcal {C}}_{sep, 2}(\Sigma _g)$ is the simplicial graph where vertices correspond to (isotopy classes of) separating curves c bounding a subsurface homeomorphic to $\Sigma _{2,1}$, and where c and d are adjacent in ${\mathcal {C}}_{sep,2}(\Sigma _g)$ whenever c and d are disjoint in $\Sigma _g$.

Lemma 7.3

${\mathcal {C}}_{sep,2}(\Sigma _g)$ is connected for $g \ge 5.$

Proof

This is a straightforward consequence of Theorem 7.1. With reference to Fig. 7 and the standard generating set of Fig. 1, observe that only the generator $T_{c_2}^{\pm }$ does not fix the base vertex c. In this case, the genus 2 subsurface determined by d is disjoint from both c and $T_{c_2}^\pm (c)$, and so there is a path $c,d,T_{c_2}^{\pm }(c)$ in ${\mathcal {C}}_{sep, 2}(\Sigma _g)$. $\square$

Definition 7.4

Let $\phi$ be a ${\mathbb {Z}}/r {\mathbb {Z}}$-valued spin structure on a surface $\Sigma _g$. The graph ${\mathcal {C}}_\phi (\Sigma _g)$ has vertices consisting of the admissible curves for $\phi$, where a and b are adjacent whenever $i(a,b) = 0$. The graph $\mathcal C_\phi ^1(\Sigma _g)$ has the same vertex set as $\mathcal C_\phi (\Sigma _g)$, but vertices a, b are adjacent whenever $i(a,b) = 1$.

Lemma 7.5

${\mathcal {C}}_\phi ^1(\Sigma _g)$ is connected for $g \ge 5$.

Fig. 7

The configuration of curves needed for Lemma 7.3

Proof

The first step is to establish the connectivity of $\mathcal C_\phi (\Sigma _g)$. Let a, b be vertices. Choose subsurfaces $S_{a}, S_{b}$ containing a, b respectively, each homeomorphic to $\Sigma _{2,1}$. By Lemma 7.3, there is a path $S_{a_0}, \dots , S_{a_n}$ in ${\mathcal {C}}_{sep,2}(\Sigma _g)$ with $a \subset S_{a_0}$ and $b\subset S_{a_n}$, with each $S_{a_i}$ disjoint from $S_{a_{i+1}}$. By Corollary 4.3, on each $S_{a_i}$ there exists some admissible curve $a_i$. By construction, $a = a_0, a_1, \dots , a_n = b$ is a path in ${\mathcal {C}}_\phi (\Sigma _g)$ connecting a to b.

The connectivity of ${\mathcal {C}}_\phi ^1(\Sigma _g)$ now follows readily. Given a path $a = a_0, \dots , a_n = b$ in $\mathcal C_\phi (\Sigma _g)$, Corollary 4.4 implies that for each i, there exists some admissible curve $c_i$ such that $i(a_i, c_i) = i(a_{i+1},c_i) = 1$. The path $a_0, c_0, a_1, c_1, \dots , c_{n-1}, a_n$ connects a to b in $\mathcal C_\phi ^1(\Sigma _g)$. $\square$

8 Subsurface push subgroups and ${\mathcal {T}}_\phi$

As discussed in the introduction, the main technical result on the groups ${{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ]$ and ${\mathcal {T}}_\phi$ that we require is a criterion for a collection of Dehn twists to generate $\mathcal T_\phi$, given below as Theorem 9.5. This is the first of two sections dedicated to proving Theorem 9.5. Here, we formulate and prove the intermediate result Proposition 8.2, which gives a generating set for ${\mathcal {T}}_\phi$not consisting entirely of Dehn twists. The results here concern a class of subgroups known as spin subsurface push subgroups; these are introduced in Sects. 8.1 and 8.2.

8.1 Subsurface push subgroups

Recall the classical inclusion map, as discussed in [7, Theorem 3.18]. Let $S' \subset S$ be a subsurface either of genus $g(S') \ge 2$ with $n \ge 1$ boundary components, or else of genus $g(S') = 1$ with $n \ge 2$ boundary components. Assume that no component of $\partial S'$ bounds a closed disk in S. Let $a_1, \dots , a_k$ denote the boundary components of $S'$ that bound punctured disks in S, let $b_1, b_1', \dots , b_\ell , b_\ell '$ denote the pairs of boundary components of $S'$ that cobound an annulus in S, and $c_1, \dots , c_m$ denote the remaining boundary components. Let $i_*{:}\,{{\,\mathrm{Mod}\,}}(S') \rightarrow {{\,\mathrm{Mod}\,}}(S)$ denote the map on mapping class groups arising from the inclusion $i{:}\,S' \hookrightarrow S$. Then

$$\begin{aligned} \ker (i_*) = \left\langle {T_{a_1}, \dots , T_{a_k}, T_{b_1} T_{b_1'}^{-1}, \dots , T_{b_\ell }T_{b_\ell '}^{-1}}\right\rangle . \end{aligned}$$

Let $\Delta$ be a boundary component of $S'$, and suppose that $\Delta$ does not bound a punctured disk in S. Let $\overline{S'}$ denote the surface obtained from $S'$ by capping off $\Delta$ with a closed disk. According to (2), there is a subgroup of ${{\,\mathrm{Mod}\,}}(S')$ isomorphic to $\pi _1(UT\overline{S'})$. The subsurface push subgroup for $(S', \Delta )$ is defined to be the image of $\pi _1(UT\overline{S'})$ under the inclusion $i_*{:}\,{{\,\mathrm{Mod}\,}}(S') \rightarrow {{\,\mathrm{Mod}\,}}(S)$. This will be written $\Pi (S',\Delta )$, or simply $\Pi (S')$ if the boundary component does not need to be emphasized.

We remark here that $i_*$ restricts to an injection$\pi _1(UT\overline{S'}) \hookrightarrow {{\,\mathrm{Mod}\,}}(S)$, even when there exists some other boundary component $\Delta '$ of $S'$ such that $\Delta \cup \Delta '$ cobounds an annulus on S. To see this, observe that $\pi _1(UT\overline{S'})\leqslant {{\,\mathrm{Mod}\,}}(S')$ is characterized by the property that $f \in \pi _1(UT\overline{S'})$ if and only if f becomes isotopic to the identity when extended to $\overline{S'}$. It is easy to see that no element of $\ker (i_*)$ has this property.

8.2 Spin subsurface push subgroups

Let $S' \subset S$ be a subsurface with some boundary component $\Delta$ satisfying $\phi (\Delta ) = -1$. The following lemma shows that ${{\,\mathrm{Mod}\,}}(S)[\phi ]$ contains a finite-index subgroup of $\Pi (S', \Delta )$. This subgroup, written $\widetilde{\Pi }(S', \Delta )$, is called a spin subsurface push subgroup. Before proceeding with the rest of the section, the reader may wish to review the notion of a fundamental multitwist defined in Sect. 3.3.

Lemma 8.1

Let $S' \subset S$ be a subsurface with some boundary component $\Delta$ satisfying $\phi (\Delta ) = -1$. Then there is a finite-index subgroup ${\widetilde{\Pi }}(S', \Delta ) \leqslant {{\,\mathrm{Mod}\,}}(S)[\phi ] \cap \Pi (S', \Delta )$ characterized by the diagram given below, whose rows are short exact sequences:

(10)

The subgroup ${\widetilde{\Pi }}(S', \Delta )$ contains all fundamental multitwists for pairs of pants $P \subset S'$ of the form $P = a \cup b \cup \Delta$.

Proof

Following the discussion of Sect. 2.2, there exists a “geometric” generating set for $\pi _1(UT\overline{S'}, \Delta )$ of the following form:

$$\begin{aligned} \pi _1(UT\overline{S'})= \langle {\tilde{\alpha }}_1, \dots , \tilde{\alpha }_k, \zeta \rangle . \end{aligned}$$

(11)

Here $\alpha _i$ is some simple closed curve on $\overline{S'}$ based at $\Delta$, and ${\tilde{\alpha }}_i$ denotes the Johnson lift to $\pi _1(UT\overline{S'})$. As before, $\zeta$ denotes the loop around the fiber. As an element of ${{\,\mathrm{Mod}\,}}(S')$, each ${\tilde{\alpha }}_i$ is of the form $T_{\alpha _{i,L}} T_{\alpha _{i,R}}$, where $\alpha _{i,L}$ denotes the curve on $S'$ lying to the left of $\alpha _i$ and $\alpha _{i,R}$ lies to the right. It follows that $P_i = \alpha _{i,L} \cup \alpha _{i,R} \cup \Delta$ forms a pair of pants on $S'$. Following Lemma 3.18, the fundamental multitwist

$$\begin{aligned} T_{P_i} = T_{\alpha _{i,L}} T_{\alpha _{i,R}}^{-1} T_\Delta ^{\phi (\alpha _{i,R})} \end{aligned}$$

lies in ${{\,\mathrm{Mod}\,}}(S)[\phi ] \cap \Pi (S', \Delta )$. Embedding $\pi _1(UT\overline{S'})$ into ${{\,\mathrm{Mod}\,}}(S')$, the generating set of (11) can be replaced by the following generating set for $\Pi (S', \Delta )$:

$$\begin{aligned} \Pi (S', \Delta ) = \langle T_{P_1}, \dots , T_{P_k}, T_\Delta \rangle . \end{aligned}$$

Define

$$\begin{aligned} \widetilde{\Pi }(S', \Delta ) = \langle T_{P_1}, \dots , T_{P_k}, T_\Delta ^r \rangle . \end{aligned}$$

By construction, $\widetilde{\Pi }(S', \Delta ) \leqslant {{\,\mathrm{Mod}\,}}(S)[\phi ]$. Under the projection $\Pi (S', \Delta ) \rightarrow \pi _1(\overline{S'})$, the set $\{T_{P_i}\}$ maps onto a generating set for $\pi _1(\overline{S'})$. It follows that ${\widetilde{\Pi }}(S', \Delta )$ surjects onto $\pi _1(\overline{S'})$. As $T_\Delta ^m \in {{\,\mathrm{Mod}\,}}(S)[\phi ]$ if and only if $r \mid m$, it follows that ${\widetilde{\Pi }}(S', \Delta )$ is indeed characterized by the diagram (10) as claimed.

For the second claim, let $P = a \cup b \cup \Delta$ be a pair of pants on $S'$. The curves a, b are isotopic on $\overline{S'}$ and cobound an annulus containing the basepoint. It follows that $T_a T_b^{-1} \in \pi _1(\overline{S'})$. Via (10), there is some lift $T_a T_b^{-1} T_\Delta ^k \in \widetilde{\Pi }(S',\Delta )$, and as $T_\Delta ^r \in {\widetilde{\Pi }}(S', \Delta )$ as well, it follows that all fundamental multitwists for P are elements of ${\widetilde{\Pi }}(S', \Delta )$ as claimed. $\square$

For the purposes of this paper, we will most often be concerned with subsurface push subgroups for a special class of subsurfaces. Let $b \subset \Sigma _g$ be a nonseparating closed curve satisfying $\phi (b) = -1$. The boundary component $\Delta$ of $\Sigma _g {\setminus }\{b\}$ corresponding to the left side of b satisfies $\phi (\Delta ) = -1$, and to ease notation, we write $\widetilde{\Pi }(\Sigma _g {\setminus }\{b\})$ to refer to this spin subsurface push subgroup.

8.3 Generating admissible twists

We have arrived at the key result of the section.

Proposition 8.2

Let $\phi$ be a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure on a closed surface $\Sigma _g$ for $g \ge 5$ and any integer r. Let $(a_0,a_1,b)$ be an ordered 3-chain of curves with $\phi (a_0) = \phi (a_1) = 0$ and $\phi (b) = -1$. Let $H \leqslant {{\,\mathrm{Mod}\,}}(\Sigma _g)$ be a subgroup containing $T_{a_0}, T_{a_1}$ and the spin subsurface push group ${\widetilde{\Pi }}(\Sigma _g {\setminus }\{b\})$. Then H contains ${\mathcal {T}}_\phi$.

The proof will require the preliminary Lemma 8.3, for which we introduce some terminology. For a subgroup $H \leqslant {{\,\mathrm{Mod}\,}}(\Sigma _g)$, we say that a simple closed curve a is an H-curve if $T_a \in H$. We also say that curves a, b are H-equivalent if there exists some $f \in H$ with $f(a) = b$. If a and $b = f(a)$ are H-equivalent and ${\widetilde{\Pi }}(\Sigma _g {\setminus }\{a\}) \leqslant H$, then also ${\widetilde{\Pi }}(\Sigma _g{\setminus }\{b\}) = f\widetilde{\Pi }(\Sigma _g{\setminus }\{a\}) f^{-1}$ is a subgroup of H.

The following lemma establishes some sufficient conditions for H-equivalence of curves.

Lemma 8.3

Let $\Sigma _g$ be a surface of genus $g \ge 5$. Let $a_0,a_1,b$ be an ordered 3-chain of curves with $\phi (a_0) = \phi (a_1) = 0$ and $\phi (b) = -1$. Let $H \leqslant {{\,\mathrm{Mod}\,}}(\Sigma _g)$ be a subgroup containing $T_{a_0}, T_{a_1}$ and ${\widetilde{\Pi }}(\Sigma _g {\setminus } \{b\})$.

• (1) Let $b'$ be an oriented curve satisfying $\phi (b') = -1$ such that $i(b, b') = 0$ and $i(a_1, b') = 1$. Then b and $b'$ are H-equivalent. It follows that ${\widetilde{\Pi }}(\Sigma _g {\setminus } \{b'\}) \leqslant H$.

• (2) Let $a'$ be any nonseparating curve satisfying $\phi (a')= 0$ such that $a'$ is disjoint from the configuration $a_0 \cup a_1 \cup b$. Then $a'$ is an H-curve.

• (3) Let $b'$ be any nonseparating curve satisfying $\phi (b') = -1$ such that $b'$ is disjoint from the configuration $a_0 \cup a_1 \cup b$. Then b and $b'$ are H-equivalent, and hence ${\widetilde{\Pi }}(\Sigma _g {\setminus } \{b'\}) \leqslant H$.

Proof

• (1) If $b = b'$ there is nothing to prove. Otherwise, given $a_1, b, b'$, we define a curve $b''$ as follows. Let $\epsilon$ be the portion of $a_1$ connecting the left side of b to one of the sides of $b'$; then $b''$ is defined as the curve-arc sum $b'' \,{:}{=}\, b +_\epsilon b'$. By construction $b\cup b' \cup b''$ bounds a pair of pants P lying to the left of b, and $i(a_1, b'') = 0$. By Lemma 3.13, there exists an orientation of $b''$ such that $\phi (b'') = -1$. This can be determined as follows: $b''$ is oriented with P lying to the right if and only if P lies to the left of $b'$. If P lies to the left of $b'$, then the element $T_b T_{b'} T_{b''}^{-1}$ is a fundamental multitwist and hence an element of ${\widetilde{\Pi }}(\Sigma _g {\setminus }\{b\}) \leqslant H$. Otherwise, $T_b T_{b'}^{-1} T_{b''}$ is a fundamental multitwist. In the first case, the braid relation implies that

  $$\begin{aligned} T_{a_1}(T_b T_{b'} T_{b''}^{-1})T_{a_1}(b) = b', \end{aligned}$$

  while in the second case,

  $$\begin{aligned} T_{a_1}^{-1}(T_b T_{b'}^{-1} T_{b''})T_{a_1}(b) = b'. \end{aligned}$$

  In either case, the indicated element lies in H, showing the H-equivalence between $b, b'$.

• (2) Let $\epsilon$ be an arc connecting $a_0$ to $a'$ that is disjoint from $a_1 \cup b$, and define $b' \,{:}{=}\, a_0+_\epsilon a'$. It is possible that $b' = b$, but this will not pose any difficulty. Then $a_0 \cup a' \cup b'$ forms a pair of pants and $b'$ satisfies the intersection conditions $i(b,b') = 0$ and $i(a_1, b') = 1$. By the homological coherence property, $\phi (b') = -1$. By the second assertion of (1), ${\widetilde{\Pi }}(\Sigma _g {\setminus } \{b'\}) \leqslant H$. As $a_0 \cup a' \cup b'$ forms a pair of pants, it follows that $T_{a_0} T_{a'}^{-1}$ is a fundamental multitwist, and so $T_{a_0}T_{a'}^{-1} \in {\widetilde{\Pi }}(\Sigma _g {\setminus } \{b'\}) \leqslant H$. As $T_{a_0} \in H$ by hypothesis, this shows that $T_{a'} \in H$ as desired.

• (3) Given $b'$, Corollary 4.4 implies that there exists an admissible curve $a'$ that is disjoint from $a_0 \cup a_1 \cup b$ and for which $i(a',b') = 1$. Corollary 4.4 also establishes the existence of a curve $b''$, satsifying $\phi (b'') = -1$, with the following intersection properties:

  $$\begin{aligned} i(b, b'') = i(b', b'') = i(a_0, b'') = 0; \qquad i(a', b'') = i(a_1, b'') = 1. \end{aligned}$$

  By (1), b and $b''$ are H-equivalent. By (2), $a'$ is an H-curve, so that by (1) again, $b''$ and $b'$ are H-equivalent, showing the result. $\square$

Proof of Proposition 8.2

Let a be any admissible curve. There is some genus 2 subsurface $S' \cong \Sigma _{2,1}$ containing a, and there is also some genus 2 subsurface $S \cong \Sigma _{2,1}$ that contains the curves $a_0,a_1, b$. By Lemma 7.3, there is a path $S_0 = S - S_1 - \dots - S_n = S'$ of subsurfaces homeomorphic to $\Sigma _{2,1}$ with boundary components $\partial S_i$ and $\partial S_{i+1}$ disjoint for $i = 1, \dots , n-1$, hence $S_i \cap S_{i+1} = \emptyset$ for $i = 1, \dots , n-1$.

For $i = 1, \dots , n$, let $a_{2i}$ be an admissible curve contained in $S_i$; we take $a_{2n} = a$. We claim that there exist curves $a_{2i+1}$ and $b_i$ on $S_i$ such that $a_{2i}, a_{2i+1}, b_i$ forms a chain, and $\phi (a_{2i+1}) = 0, \phi (b_i) = -1$. To see this, let $T \subset S_i$ be a subsurface of genus 1 that does not contain $a_{2i}$. By Corollary 4.3, there is an admissible curve $a'$ contained in T. Let $\epsilon$ be an arc connecting $a_{2i}$ and $a'$; then $b_i\,{:}{=}\, a_{2i} +_\epsilon a'$ satisfies $\phi (b) = -1$ for a suitable choice of orientation. Let c be any curve on $S_i$ such that $i(c,a_{2i}) = i(c, b_i) = 1$. Then $a_{2i+1} \,{:}{=}\, T_{b_i}^{\phi (c)}(c)$ is admissible, and $a_{2i}, a_{2i+1}, b_i$ forms a chain as required.

We assume for the sake of induction that $a_{2i}, a_{2i+1}$ are H-curves and that ${\widetilde{\Pi }}(\Sigma _{g} {\setminus } \{b_i\}) \leqslant H$. Then by Lemma 8.3.2, also $a_{2i+2}, a_{2i+3}$ are H-curves, and ${\widetilde{\Pi }}(\Sigma _{g} {\setminus } \{b_{i+1}\}) \leqslant H$. The base case $i = 0$ holds by hypothesis, taking $b_0 = b$. The claim now follows by induction. $\square$

9 Networks

In this section we deduce Theorem 9.5 from Proposition 8.2. The key notion is that of a network of curves. In Sect. 9.1, we establish the basic theory of networks, and in Sect. 9.2, we state and prove Theorem 9.5. Departing from our conventions elsewhere in the paper, in this section we work with individual curves and not merely their isotopy classes.

9.1 Networks and their basic theory

Definition 9.1

Let $S = \Sigma _{g,b}^n$ be a surface, viewed as a compact surface with marked points. A network on S is any collection ${\mathcal {N}} = \{a_1,\dots , a_n\}$ of simple closed curves on S, disjoint from any marked points, such that $\#(a_i \cap a_j) \le 1$ for all pairs of curves $a_i, a_j \in {\mathcal {N}}$, and such that there are no triple intersections. A network ${\mathcal {N}}$ has an associated intersection graph$\Gamma _{{\mathcal {N}}}$, whose vertices correspond to curves $x \in {\mathcal {N}}$, with vertices x, y adjacent if and only if $\#(x\cap y) = 1$. A network is said to be connected if $\Gamma _{{\mathcal {N}}}$ is connected, and arboreal if $\Gamma _{{\mathcal {N}}}$ is a tree. A network is filling if

$$\begin{aligned} S {\setminus } \bigcup _{a \in {\mathcal {N}}} a \end{aligned}$$

is a disjoint union of disks and boundary-parallel annuli; each component is allowed to contain at most one marked point of S.

The data of a network encodes both an abstract finite set of curves as well as a topological subspace of the surface S. To avoid confusing these, let the symbol ${\mathcal {N}}$ denote this finite set, and let $\widehat{{\mathcal {N}}}$ denote the space. When ${\mathcal {N}}$ is arboreal, there is a simple generating set for $\pi _1(\widehat{{\mathcal {N}}})$. To describe it, endow $\widehat{{\mathcal {N}}}$ with the structure of a CW complex, and let ${\mathcal {T}}$ be a spanning tree for this CW complex.

Lemma 9.2

Let ${\mathcal {N}}$ be an arboreal network. Then there is a 1–1 correspondence between the set of edges $\widehat{\mathcal N}{\setminus } {\mathcal {T}}$, and the set ${\mathcal {N}}$.

Proof

Each edge of $\widehat{{\mathcal {N}}}$ is contained in a unique element of ${\mathcal {N}}$. For a given $a \in {\mathcal {N}}$, let $a_1, \dots , a_{n(a)}$ denote these edges, ordered so that adjacent edges are numbered consecutively. For each $a \in {\mathcal {N}}$, the sequence $a_1, \dots , a_{n(a)}$ forms a cycle in $\widehat{{\mathcal {N}}}$. Thus for each $a \in {\mathcal {N}}$, there is at least one edge $a_1$ (without loss of generality) that is not contained in ${\mathcal {T}}$.

It remains to show that for each $a \in {\mathcal {N}}$, there is exactly one edge not contained in ${\mathcal {T}}$. Equivalently, we must show that the intersection $a \cap {\mathcal {T}}$ is connected as a topological space. The assumption that ${\mathcal {N}}$ is arboreal implies that $\Gamma _{{\mathcal {N}}}$ has the following property: let $\Gamma _{{\mathcal {N}}}(a)$ be the graph obtained from $\Gamma _{{\mathcal {N}}}$ by removing all edges incident to a. Then each vertex b adjacent to a in $\Gamma _{{\mathcal {N}}}$ determines a distinct component of $\Gamma _{{\mathcal {N}}}(a)$.

Let $v,w \in a$ be vertices of $\widehat{{\mathcal {N}}}$, and let $e_1, \dots , e_n$ be the unique geodesic path in ${\mathcal {T}}$ connecting v to w. It suffices to show that this path is contained in a. If this is not the case, let $k_1$ (resp. $k_2$) be the minimal (resp. maximal) integer such that $e_{k_1}$ (resp. $e_{k_2}$) is not contained in a. Then exactly one vertex $v_1$ of $e_{k_1}$ (resp. $v_2$ of $e_{k_2}$) lies on a, and the other lies on some adjacent curve $b_1$ (resp. $b_2$). As $i(a,b_1) = i(a,b_2) =1$ and the path $e_1, \dots , e_n$ visits each vertex in $\widehat{{\mathcal {N}}}$ at most once, it follows that $b_1$ and $b_2$ are distinct elements of ${\mathcal {N}}$. As explained in the above paragraph, the arboreality assumption implies that every path in $\widehat{{\mathcal {N}}}$ connecting a point in $b_1$ to a point in $b_2$ must pass through a. Any such path must pass through $v_1$ and $v_2$: this shows that if the path $e_1, \dots , e_n$ enters $b_1$, it must pass through $v_1$ at least twice, contrary to assumption. $\square$

Via Lemma 9.2, each $a \in {\mathcal {N}}$ determines a unique based loop P(a) by following the unique path in ${\mathcal {T}}$ from the basepoint to a. Lemma 9.3 below now follows by a standard application of the Seifert–Van Kampen theorem.

Lemma 9.3

Let ${\mathcal {N}}$ be an arboreal network. Then $\pi _1(\widehat{{\mathcal {N}}})$ is generated by the set of loops $\{P(a) \mid a \in {\mathcal {N}}\}$. If ${\mathcal {N}}$ is moreover filling, then the map $\pi _1(\widehat{{\mathcal {N}}}) \rightarrow \pi _1(S)$ is a surjection, and so $\pi _1(S)$ is also generated by this collection of loops.

$\pi _1(S)$ is a normal subgroup of ${{\,\mathrm{Mod}\,}}(S)$: if $\alpha \in \pi _1(S)$ is a mapping class corresponding to a based loop and $f \in {{\,\mathrm{Mod}\,}}(S)$ is arbitrary, then conjugation by f takes $\alpha$ to the mapping class corresponding to the based loop $f(\alpha )$. In the context of the “network presentation” of $\pi _1(S)$ arising from the surjection $\pi _1(\widehat{{\mathcal {N}}}) \rightarrow \pi _1(S)$, this means that $\pi _1(S)$ has a very simple normal generating set as a subgroup of ${{\,\mathrm{Mod}\,}}(S)$, as the following makes precise.

Lemma 9.4

Let $\widehat{{\mathcal {N}}} \subset S$ be an arboreal filling network. Let $H \leqslant {{\,\mathrm{Mod}\,}}(S)$ be a subgroup containing $T_a$ for each $a \in {\mathcal {N}}$. If H also contains $P(a_1) \in \pi _1(S)$ for some $a_1 \in {\mathcal {N}}$, then H contains the entire point-pushing subgroup $\pi _1(S)$.

Proof

As recorded in Lemma 9.3, $\pi _1(\widehat{\mathcal N})$, and hence also $\pi _1(S)$, is generated by the collection of elements P(a) for $a \in {\mathcal {N}}$. We will proceed by induction. Define connected subnetworks

$$\begin{aligned} {\mathcal {N}}_0 \subset {\mathcal {N}}_1 \subset \cdots \subset {\mathcal {N}}_n = {\mathcal {N}} \end{aligned}$$

as follows: ${\mathcal {N}}_k$ consists of all those curves a at a distance of at most k from the base vertex $a_1 \in \Gamma _{{\mathcal {N}}}$ (viewing $\Gamma _{{\mathcal {N}}}$ as a metric space for which each edge has length 1). We suppose that $\pi _1(\widehat{{\mathcal {N}}}_k) \leqslant H$; the base case $k = 0$ holds by hypothesis.

Let $a \in {\mathcal {N}}_{k+1}{\setminus } {\mathcal {N}}_k$ be arbitrary. Let $a' \in {\mathcal {N}}_k$ be adjacent to a. By the braid relation,

$$\begin{aligned} T_{a'} T_{a} (a') = a, \end{aligned}$$

and hence $P(a) = (T_{a'} T_{a}) P(a') (T_{a'} T_{a}) ^{-1} \in H$. This completes the inductive step. $\square$

9.2 Network generating sets for ${\mathcal {T}}_\phi$

Having established some of the basic theory of networks, we can now formulate and prove the key technical result of the paper. For hypotheses (2) and (3), the reader may wish to consult Fig. 2 and the surrounding discussion of the $D_n$ relation (Proposition 2.6) and the configuration ${\mathscr {D}}_n$. For an example of a network satisfying the hypotheses of Theorem 9.5, see Fig. 8.

Theorem 9.5

Let $\phi$ be a ${\mathbb {Z}}/r{\mathbb {Z}}$-valued spin structure on a closed surface $\Sigma _g$, with $1 \le r < g - 1$. Let ${\mathcal {N}} = \{a_n\}$ be a connected filling network of curves on $\Sigma _g$ with the following properties:

• (1) Every element $a_n$ is admissible,

• (2) There is a collection $a_1, \dots , a_{2r + 4}$ of elements of ${\mathcal {N}}$ such that $a_1, \dots , a_{2r + 3}$ are arranged in the configuration of the curves ${\mathscr {D}}_{2r+3}$ of the $D_{2 r + 3}$ relation, and $a_{2r + 4}$ corresponds to the boundary component $\Delta _1$ associated to the subconfiguration ${\mathscr {D}}_{2 r + 2}$.

• (3) Let $b \subset \Sigma _g$ correspond to the curve $\Delta _0$ of the $D_{2 r + 3}$ relation, as appearing in Fig. 2. Then ${\mathcal {N}}$ must contain some curve d with $i(d,b) = 1$.

• (4) Let ${\mathcal {N}}' \subset {\mathcal {N}}$ be the subnetwork consisting of all curves in ${\mathcal {N}}$ disjoint from b. Then ${\mathcal {N}}'$ must be an arboreal filling network for $\Sigma _g {\setminus } \{b\}$.

If $g \ge 5$, then $\langle T_{a_i},\ a_i \in {\mathcal {N}} \rangle$ contains the admissible subgroup ${\mathcal {T}}_\phi$.

Moreover, if r is odd, then

$$\begin{aligned} \langle T_{a_i},\ a_i \in {\mathcal {N}} \rangle = {{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ]. \end{aligned}$$

If $r=2d$ is even and $g \ge g(r)$ for the function g(r) of Definition 6.1, then $\langle T_{a_i},\ a_i \in {\mathcal {N}} \rangle$ is a subgroup of finite index in ${{\,\mathrm{Mod}\,}}(\Sigma _g)[\phi ]$.

Proof

Define

$$\begin{aligned} H = \langle T_{a_i},\ a_i \in {\mathcal {N}} \rangle . \end{aligned}$$

By hypothesis (1), $H \leqslant {\mathcal {T}}_\phi$. We establish the opposite containment ${\mathcal {T}}_\phi \leqslant H$. The remaining assertions in the statement of Theorem 9.5 follow by an appeal to Propositions 5.1 or 6.2 as appropriate. The containment ${\mathcal {T}}_\phi \leqslant H$ will follow from Proposition 8.2. To see that the hypotheses of Proposition 8.2 are satisfied by H, it is necessary to establish a containment ${\widetilde{\Pi }}(\Sigma _g {\setminus }\{b\})\leqslant H$, and to find suitable curves corresponding to $a_0, a_1$ in the statement of Proposition 8.2.

Consider the curves $\{a_1, \dots , a_{2r + 4}\} \subset {\mathcal {N}}$ corresponding to ${\mathscr {D}}_{2 r + 3} \cup \{\Delta _1\}$ as in Corollary 2.7, as posited by hypothesis (2). Without loss of generality, assume that $a_1, a_2, a_3 \in {\mathcal {N}}$ correspond to the curves $a, c_1, a'$ of ${\mathscr {D}}_{2 r +3}$, so that $b \subset \Sigma _g$ is one of the boundary components of the chain $a_1, a_2, a_3$. Let d be the curve with $i(d,b) = 1$ posited by hypothesis (3), and let P be the pair of pants bounded by $a_1, a_3, b$. The intersection $d \cap P$ must be a single arc, since $d \in {\mathcal {N}}$ and so $\#(d\cap a_1) \le 1$ and $\#(d \cap a_3) \le 1$. Without loss of generality, assume $\#(d\cap a_1) = 1$ and $\#(d \cap a_3) = 0$. Then the 3-chain $a_1, d, b$ on $\Sigma _g$ corresponds to the 3-chain $a_0, a_1, b$ of Proposition 8.2, since $\phi (b) = -1$ by the homological coherence property. By assumption, $T_d, T_{a_1} \in H$, so it remains only to establish ${\widetilde{\Pi }}(\Sigma _g {\setminus }\{b\})\leqslant H$.

By hypothesis (4), the restriction ${\mathcal {N}}'$ determines an arboreal filling network on $\Sigma _g{\setminus }\{b\}$. The same is therefore true on the surface $\overline{\Sigma _g {\setminus } \{b\}}$ obtained by filling in the boundary component corresponding to the left side of b (where b is oriented so that $a_1, a_3$ lie to the left). The surface $\overline{\Sigma _g {\setminus } \{b\}}$ is connected, since the hypothesis $i(d,b) =1$ implies that d is nonseparating. We treat $\overline{\Sigma _g {\setminus } \{b\}}$ as a surface $\Sigma _{g-1, 1}^1$, with the marked point corresponding to the filled-in left side of b.

We will show that ${\widetilde{\Pi }}(\Sigma _g {\setminus } \{b\}) \leqslant H$ by appealing to Lemma 8.1. We must therefore show that $T_b^r$ is an element of H, and show that the image of the map

$$\begin{aligned} \langle T_a, a \in {\mathcal {N}}' \rangle \rightarrow {{\,\mathrm{Mod}\,}}(\overline{\Sigma _g {\setminus } \{b\}}) \end{aligned}$$

contains the point-pushing subgroup $\pi _1(\overline{\Sigma _g {\setminus } \{b\}})$. Applying Corollary 2.7.2, we obtain $T_b^r \in H$. To exhibit $\pi _1(\overline{\Sigma _g {\setminus } \{b\}})$, we will appeal to Lemma 9.4. The element $T_{a_1} T_{a_3}^{-1}$ corresponds to an element $P(a_1) \in \pi _1(\overline{\Sigma _g {\setminus } \{b\}})$. By Lemma 9.4, it follows that the entire point-pushing subgroup $\pi _1(\overline{\Sigma _g {\setminus } \{b\}})$ is contained in the subgroup $\langle T_a, a \in {\mathcal {N}}' \rangle \leqslant H$. $\square$

10 Linear systems in toric surfaces

The purpose of this section is to give a minimal account of the work of Crétois–Lang in [3]. We do not attempt to give a detailed summary of the theory of toric surfaces; the interested reader is referred to [3] and the references therein.

Consider the integer lattice ${\mathbb {Z}}^2 \subset {\mathbb {R}}^2$. A lattice polygon$\Delta$ is the convex hull of a finite collection $\{v_1, \dots , v_n\}$ of $n \ge 3$ elements $v_i \in {\mathbb {Z}}^2$, not all collinear. Given a polygon $\Delta$ which contains at least one lattice point in the interior ${\text {int}}(\Delta )$, the adjoint polygon$\Delta _a$ is defined to be the convex hull of ${\text {int}}(\Delta ) \cap {\mathbb {Z}}^2$.

The following proposition is a concise summary of the correspondence between line bundles on toric surfaces and polygons. For details, see [3, Section 3]. In item (1), a unimodular transformation of ${\mathbb {R}}^2$ is an affine map $A{:}\,{\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2$ (necessarily invertible) such that $A {\mathbb {Z}}^2 = {\mathbb {Z}}^2$.

Proposition 10.1

Let X be a smooth toric surface.

• (1) Associated to any nef line bundle ${\mathcal {L}}$ on X is a convex lattice polygon $\Delta _{{\mathcal {L}}}$, well-defined up to unimodular transformations.

• (2) If ${\mathcal {L}}$ is nef, then the roots of ${\mathcal {L}}$ (i.e. the line bundles ${\mathcal {S}}$ for which $n{\mathcal {S}} = {\mathcal {L}}$ for some integer n) are in correspondence with the dilates $\frac{1}{n} \Delta _{{\mathcal {L}}}$ for which $\frac{1}{n} \Delta _{{\mathcal {L}}}$ is a lattice polygon.

• (3) Suppose that ${\mathcal {L}}$ is ample and that ${\text {int}}(\Delta _{{\mathcal {L}}}) \cap {\mathbb {Z}}^2$ is nonempty. Then the adjoint line bundle ${\mathcal {L}} \otimes K_X$ is nef, and $\Delta _{{\mathcal {L}} \otimes K_X} = (\Delta _{{\mathcal {L}}})_a$.

• (4) Let ${\mathcal {L}}$ be ample. The genus $g({\mathcal {L}})$ of a smooth $C \in \left| {\mathcal {L}} \right|$ is given by the formula

  $$\begin{aligned} g({\mathcal {L}}) = \# ({\text {int}}(\Delta _{{\mathcal {L}}}) \cap {\mathbb {Z}}^2) = \#((\Delta _{{\mathcal {L}}})_a \cap {\mathbb {Z}}^2). \end{aligned}$$

• (5) Let ${\mathcal {L}}$ be ample. A generic fiber $C \in \left| {\mathcal {L}} \right|$ is hyperelliptic if and only if $(\Delta _{{\mathcal {L}}})_a$ is a line segment.

The following proposition indicates the connection between the divisibility properties of ${\mathcal {L}} \otimes K_X$ as an element of ${{\,\mathrm{Pic}\,}}(X)$ (or after Proposition 10.1.2, the divisibility of $(\Delta _{\mathcal {L}})_a$), and the presence of invariant higher spin structures. It is a folklore theorem; see [17, Theorem 1.1] and [3, Proposition 2.7] for written accounts.

Proposition 10.2

Let ${\mathcal {L}}$ be an ample line bundle on a smooth toric surface X. For any r such that the adjoint line bundle ${\mathcal {L}} \otimes K_X$ admits a $r\text {th}$ root in ${{\,\mathrm{Pic}\,}}(X)$, there exists a (unique) ${\mathbb {Z}}/r {\mathbb {Z}}$-valued spin structure $\phi$ preserved by the monodromy $\mu _{\mathcal {L}}$:

$$\begin{aligned} \Gamma _{\mathcal {L}} \leqslant {{\,\mathrm{Mod}\,}}(\Sigma _{g({\mathcal {L}})})[\phi ]. \end{aligned}$$

Proposition 10.1 suggests that it might be profitable to “model” a smooth $C \in \left| {\mathcal {L}} \right|$ on the lattice polygon $\Delta _{\mathcal {L}}$.

Construction 10.3

(Inflation procedure) Let $\Delta$ be a lattice polygon. Let B(r, x) denote the open ball of radius r centered at $x \in {\mathbb {R}}^2$. Define the surface with boundary

$$\begin{aligned} \Delta ^\circ \,{:}{=}\, \Delta {\setminus } \bigcup _{v \in {\text {int}}(\Delta ) \cap {\mathbb {Z}}^2} B(v, 1/4). \end{aligned}$$

The inflation of $\Delta$ is the surface $C_{\Delta }$ obtained as the double of $\Delta ^\circ$ along its boundary. It is a closed oriented surface of genus $g = \# ({\text {int}}(\Delta ) \cap {\mathbb {Z}}^2)$. In particular, for $\Delta = \Delta _{{\mathcal {L}}}$ for some ample ${\mathcal {L}}$, the inflation $C_\Delta$ has genus $g(\mathcal L)$. See Fig. 8 for the example of $\mathcal O(6)$ on ${{\mathbb {CP}}}^2$.

The first indication of the utility of the inflation procedure is provided by the following theorem of Crétois–Lang. For an inflation $C_\Delta$, define an A-curve to be any simple closed curve on $C_\Delta$ that corresponds to the circle of radius 1 / 4 centered at an interior lattice point of $\Delta$.

Theorem 10.4

[3, Theorem 3] Let ${\mathcal {L}}$ be an ample line bundle on a smooth toric surface X. There is a homeomorphism $f{:}\,C_0 \rightarrow C_{\Delta _{{\mathcal {L}}}}$ identifying a smooth $C_0 \in \left| {\mathcal {L}} \right|$ with $C_{\Delta _{{\mathcal {L}}}}$, such that every A-curve $a \subset C_{\Delta _{{\mathcal {L}}}}$ is a vanishing cycle, and

$$\begin{aligned} T_a \in \Gamma _{{\mathcal {L}}}. \end{aligned}$$

Crétois–Lang also determine a second family of elements of $\Gamma _{{\mathcal {L}}}$ arising from the combinatorics of $\Delta$. A primitive integer segment is a line segment $\sigma \subset {\mathbb {R}}^2$ whose endpoints lie on ${\mathbb {Z}}^2$ and whose interior is disjoint from ${\mathbb {Z}}^2$. A primitive integer segment determines a line in ${\mathbb {R}}^2$ in the obvious way. When a lattice polygon $\Delta$ is fixed, it will be understood that a primitive integer segment connects lattice points $v,w \in \Delta \cap {\mathbb {Z}}^2$, and such that v and w do not lie along the same edge of $\Delta$. Under the inflation procedure, a primitive integer segment corresponds to a simple closed curve. For a primitive integer segment $\sigma$, we write $T_\sigma$ for the corresponding Dehn twist.

Suppose that $\Delta$ is a lattice polygon, let $d \ge 1$ be an integer. We say that $\Delta$ is divisible by d if the dilate $\frac{1}{d} \Delta$ is again a lattice polygon. If $\Delta$ is divisible by d, then after translating $\Delta$ so that one vertex lies in the sublattice $d{\mathbb {Z}}^2$, the remaining vertices do also. We write

$$\begin{aligned} \Delta (d) \,{:}{=}\, \Delta \cap d {\mathbb {Z}}^2, \end{aligned}$$

relative to any such embedding. The following is a combination of Propositions 7.13 and 7.16 of [3].

Theorem 10.5

(Crétois–Lang) Let ${\mathcal {L}}$ be an ample line bundle on a smooth toric surface X. Suppose that the adjoint polygon $(\Delta _{\mathcal {L}})_a$ is divisible by d. Suppose that $\sigma$ is a primitive integer segment such that the line it generates intersects $(\Delta _\mathcal L)_a(d)$. Then, with respect to the identification $f{:}\,C_0 \rightarrow C_{\Delta _{\mathcal {L}}}$ of Theorem 10.4, we have that $\sigma$ is a vanishing cycle and $T_\sigma \in \Gamma _\mathcal L$.

Taken together, Theorems 10.4 and 10.5 produce a large family of Dehn twists in $\Gamma _{\mathcal {L}}$. In the next section, we will see that they provide sufficiently many elements to satisfy the hypotheses of Theorem 9.5, which will lead to a proof of Theorem A.

11 Proof of Theorem A

Fix a toric surface X and an ample line bundle ${\mathcal {L}}$. This determines the polygons $\Delta _{\mathcal {L}}$ and $(\Delta _\mathcal L)_a$, as well as the monodromy group $\Gamma _{\mathcal {L}}$. For convenience, we will drop reference to ${\mathcal {L}}$ from the notation, and speak of $\Delta , \Delta _a, \Gamma$, etc. We also shorten notation for the inflation curve $C_{\Delta _{\mathcal {L}}}$, and refer simply to C instead.

By hypothesis, r is the highest root of the line bundle $\mathcal L \otimes K_X$. Proposition 10.1.2 implies that $\Delta _a$ is r-divisible. Our first objective is to find a network ${\mathcal {N}}$ satisfying the hypotheses of Theorem 9.5. This will show all but the last assertion of Theorem A. Once this has been accomplished, we will see that the answer to Question 1.2 readily follows.

Genus hypotheses We first address the genus assumptions of Theorem 9.5. Recalling that $\Delta _a$ is assumed to be r-divisible, a calculation using Pick’s formula implies that for $r > 1$,

$$\begin{aligned} g \ge \frac{(r+1)(r+2)}{2}. \end{aligned}$$

This shows that $g \ge 5$ for all $r>1$ and that $g \ge r + 1$ for $r = 2d$ even. For $r = 2d = 4$, this gives $g \ge 15$, and for $r = 2d = 8$ this gives $g \ge 45$. In all cases, the hypothesis $g \ge g(r)$ of Theorem 9.5 holds.

The remaining assumption to be addressed is the requirement that $r < g -1$. As noted in Remark 3.6, r must divide $2g -2$, so we must only show that the cases $r = 2g-2$ and $r = g-1$ do not occur in the study of linear systems on toric surfaces. Suppose first that $r = 2g-2$. This implies that the adjoint polygon $\Delta _a$ contains precisely g lattice points, but is also $2g-2$-divisible. This is an absurdity: let e be an edge of the lattice polygon $\frac{1}{2g-2} \Delta _a$; then the dilate $(2g-2)e$ contains at least $2g-1 > g$ lattice points. In the case $r = g -1$, a similar analysis shows that in fact $\Delta _a$ must equal the $g-1$-fold dilation of a primitive integer segment. By Proposition 10.1.5, this implies that the general fiber of the linear system is hyperelliptic, which we have excluded from consideration.

Constructing the network ${{\mathcal {N}}}$ Recall that according to Theorem 10.4, each integer point $v \in \Delta _a \cap {\mathbb {Z}}^2$ determines a vanishing cycle in $\Gamma$; we introduce the notation A(v) to refer to the curve associated to v. When we have a specific identification of $\Delta$ with a lattice polygon, we will use the notation A(x, y) to refer to the A-curve at the integer point (x, y). Similarly, given a primitive integer segment $\sigma$, we let $B(\sigma )$ denote the associated simple closed curve on C. When $\Delta$ is identified with a lattice polygon, we write B((x, y), (z, w)) for the B-curve associated to the primitive integer segment connecting (x, y) and (z, w). We refer to these as A-curves and B-curves, respectively.

To define the network ${\mathcal {N}}$, it will be useful to introduce some terminology. Let $\sigma$ be a primitive integer segment, and let $L(\sigma )$ be the line determined by $\sigma$. For an integer point v, we say that $\sigma$points towards v if $v \in L(\sigma )$. We also introduce the notion of a $\kappa$-standard embedding. Let $\kappa$ be a vertex of $\Delta _a$. A $\kappa$-standard embedding is an embedding of $\Delta$ into ${\mathbb {R}}^2$ such that $\kappa$ corresponds to (0, 0) and such that the edges of $\Delta _a$ incident to $\kappa$ lie along the x and y axes. Any embedding $\Delta \subset {\mathbb {R}}^2$ can be made $\kappa$-standard by applying a suitable unimodular transformation. Following Proposition 10.1.1, we are free to apply unimodular transformations as needed.

Let $\kappa$ be a vertex of $\Delta _a$. Let ${\mathcal {N}}$ be the network consisting of the following curves:

• (1) All A-curves.

• (2) The curve $B(\sigma )$, where $\sigma$ is defined as follows. Let $\kappa '$ be a vertex of $\Delta _a$ adjacent to $\kappa$. Let $e'$ be the edge of $\Delta _a$ containing $\kappa '$ and not containing $\kappa$, and let $w \in \partial \Delta _a$ be the integer point lying on $e'$ that is connected to $\kappa '$ by a primitive integer segment $\sigma$.

• (3) The curve $B(\tau )$ defined as follows. Under a $\kappa$-standard embedding of $\Delta$, necessarily $(0,-1) \in \partial \Delta$. Since $\Delta _a$ is assumed to be d-divisible, the edge of $\Delta _a$ lying along the x-axis extends at least as far as (d, 0). We take $\tau$ to be the primitive integer segment identified with $B((d,0),(0,-1))$ in this embedding of $\Delta$.

• (4) All B-curves associated to primitive integer segments pointing towards $\kappa$, but such that the associated line does not pass through the interior of the segments $\sigma$ or $\tau$ or the primitive integer segment connecting $(-1,1)$ to (0, 1).

See Fig. 8 for a picture of ${\mathcal {N}}$ in the case of the line bundle ${\mathcal {O}}(6)$ on ${{\mathbb {CP}}}^2$.

Remark 11.1

As can be seen in Fig. 8, certain elements of ${\mathcal {N}}$ are mutually isotopic. This harmless excess is introduced only to make the definition of ${\mathcal {N}}$ more tidy.

In anticipation of an appeal to Theorem 9.5, we also define the subnetwork

$$\begin{aligned} {\mathcal {N}}' \,{:}{=}\, {\mathcal {N}} {\setminus } \{A(0,1)\}. \end{aligned}$$

(12)

Fig. 8

Example: $(X, {\mathcal {L}}) = ({{\mathbb {CP}}}^2, {\mathcal {O}}(6))$; here $r = 3$. Left: the lattice polygons $\Delta$ and $\Delta _a$ (shaded). Right: the inflation construction, and the network ${\mathcal {N}}$, depicted in both shades of blue. The curves $a_1, \dots , a_{10}$ of Theorems 9.5.2 and 9.5.3 are shown in the light shade of blue. Note the curve b (in red) is not part of the network, but does correspond to the curve b of Theorem 9.5.3 (color figure online)

First properties of${{\mathcal {N}}}$ We first claim that ${\mathcal {N}}$ is a network. Indeed, all A-curves are mutually disjoint. The set of primitive integer segments under consideration meet only at integer points in $\Delta _a$, and hence the associated B-curves are also mutually disjoint. Suppose $\sigma$ has endpoints v, w. Then $i(A(v), B(\sigma )) = i(A(w), B(\sigma )) = 1$, and $i(A(u), B(\sigma )) = 0$ for any other integer point u. Thus ${\mathcal {N}}$ is a network.

Indeed, ${\mathcal {N}}$ is a connected network, as follows from the description of $\Gamma _{{\mathcal {N}}}$ and $\Gamma _{{\mathcal {N}}'}$ given below.

Lemma 11.2

The graph $\Gamma _{{\mathcal {N}}}$ has the homotopy type of $S^1$, and $\Gamma _{{\mathcal {N}}'}$ is a tree, i.e. ${\mathcal {N}}'$ is arboreal.

Proof

We first establish that $\Gamma _{{\mathcal {N}}}$ is connected. It suffices to show that every $c \in {\mathcal {N}}$ is connected to $A(\kappa )$. We first consider the case of an A-curve A(v). If $v \in \Delta _a$ is some other integer point, there is a line segment L connecting v to $\kappa$. This decomposes as a union of primitive integer segments $\sigma _i$ based at the integer points $v_j$ lying on L. Each such segment determines a B-curve in ${\mathcal {N}}$, and it is clear that there is a path from A(v) to $A(\kappa )$ alternating between $B(\sigma _i)$ and $A(v_j)$. The argument for a B-curve (including the exceptional elements $B(\sigma )$ and $B(\tau )$) is similarly straightforward.

We next claim that the subnetwork

$$\begin{aligned} {\mathcal {N}}'' \,{:}{=}\, {\mathcal {N}} {\setminus }{B(\sigma )} \end{aligned}$$

is arboreal. The curves $B(\sigma )$ and $B(\tau )$ are the only B-curves in ${\mathcal {N}}$ that do not lie on a line passing through $\kappa$. Thus the network consisting only of curves of type (1) and (4) is arboreal by construction. As $B(\tau )$ intersects only A(d, 0), this shows that the network consisting of curves of type (1), (3), and (4) is also arboreal, but this network is ${\mathcal {N}} ''$ by definition.

The curve $B(\sigma )$ intersects only the A-curves $A(\kappa ')$ and A(w). Thus $\Gamma _{{\mathcal {N}}}$ is obtained from the tree $\Gamma _{{\mathcal {N}}''}$ by adding one new vertex that is connected to two edges, so that $\Gamma _{{\mathcal {N}}} \simeq S^1$ as claimed.

It will follow from this that ${\mathcal {N}}'$ is also arboreal. The path in $\Gamma _{{\mathcal {N}}''}$ connecting $\kappa '$ to w follows the y-axis down to $\kappa$, then proceeds out along the line connecting $\kappa$ to w; in particular, it passes through the vertex (0, 1). Thus, removing A(0, 1) to create the network ${\mathcal {N}}'$ removes the single circuit in $\Gamma _{{\mathcal {N}}}$, so that $\Gamma _{{\mathcal {N}}'}$ is a tree as claimed. $\square$

We claim that ${\mathcal {N}}$ is filling. This will be established in the next two lemmas. Recall the definition of $\Delta ^\circ$ from the definition of the inflation procedure in Construction 10.3.

Lemma 11.3

Let $S \subset \Delta$ denote the union of all primitive integer segments associated to B-curves in ${\mathcal {N}}$. Then

• (1) Each component of $\Delta ^\circ {\setminus } S$ is simply-connected.

• (2) For each component D of $\Delta ^\circ {\setminus } S$, the intersection $D \cap \partial \Delta$ has at most one component.

Proof

We begin by observing that there are homotopy equivalences $\Delta ^\circ \simeq \Delta {\setminus } (\Delta _a \cap {\mathbb {Z}}^2)$ and $\Delta ^\circ {\setminus } S \simeq \Delta {\setminus } (S \cup (\Delta _a \cap {\mathbb {Z}}^2))$. It will be tidier to work with this latter space, and so we formulate our arguments in this setting.

Embed $\Delta$ into ${\mathbb {R}}^2$ and consider S as a planar graph contained in $\Delta$. Basic properties of convexity imply that for any integer point $v \in {\mathbb {Z}}^2 \cap \Delta _a$, the line connecting v and $\kappa$ does not intersect either of $B(\sigma )$ or $B(\tau )$. Hence this line determines a union of primitive integer segments in ${\mathcal {N}}$, and upon the removal of these segments over all v, there is an equality

$$\begin{aligned} \Delta {\setminus } (S \cup ({\mathbb {Z}}^2 \cap \Delta _a)) = \Delta {\setminus } S. \end{aligned}$$

To prove (1), it therefore suffices to show that $H_1(\Delta {\setminus } S; {\mathbb {Z}}) = 0$. There is a map of pairs $f{:}\,(\Delta , \partial \Delta ) \rightarrow (S^2, *)$, where $*\in S^2$ is an arbitrary basepoint. f induces a homeomorphism

$$\begin{aligned} f{:}\,\Delta {\setminus } \partial \Delta \rightarrow S^2 {\setminus } \{*\}. \end{aligned}$$

Since the segment $B(\tau )$ (among many others) intersects $\partial \Delta$, it follows that f induces a homotopy equivalence

$$\begin{aligned} f{:}\,\Delta {\setminus } S \rightarrow S^2 {\setminus } f(S), \end{aligned}$$

and hence there is an isomorphism

$$\begin{aligned} f_*{:}\,{\tilde{H}}_1(\Delta {\setminus } S; {\mathbb {Z}}) \rightarrow {\tilde{H}}_1(S^2 {\setminus } f(S); {\mathbb {Z}}). \end{aligned}$$

By Alexander duality, ${\tilde{H}}_1(S^2 {\setminus } f(S);{\mathbb {Z}}) \cong {\tilde{H}}^0(f(S);{\mathbb {Z}}) = 0$, the latter holding because S is connected by construction. This proves (1).

For (2), consider the subconfiguration $S' \subset S$ consisting of all primitive integer segments lying along a line connecting $\kappa$ to any integer point $v \in \partial \Delta$. This provides a subdivision of $\Delta$ into convex sets, each of which has a vertex at $\kappa$. Convexity then implies that each component $D_i'$ of $\Delta {\setminus } S'$ intersects $\partial \Delta$ in at most one component. The subdivision of $\Delta$ induced by S is a refinement of that induced by $S'$. Since all segments in S that intersect $\partial \Delta$ are elements of $S'$, there is an equality

$$\begin{aligned} \partial \Delta {\setminus } (S \cap \partial \Delta ) = \partial \Delta {\setminus } (S' \cap \partial \Delta ). \end{aligned}$$

Thus each component of $\partial \Delta {\setminus } (S \cap \partial \Delta )$ corresponds to a distinct component of $\Delta {\setminus } S$, and (2) follows. $\square$

Lemma 11.4

Each component of $C {\setminus } {\mathcal {N}}$ is simply-connected, i.e. ${\mathcal {N}}$ is filling.

Proof

By construction, the “deflation” map $p{:}\,C \rightarrow \Delta$ takes components of $C{\setminus } {\mathcal {N}}$ to components of $\Delta {\setminus } S$, where S continues to denote the union of all primitive integer segments associated to B-curves in ${\mathcal {N}}$. This map on components is at most 2-to-1, and is exactly 2-to-1 in the case where the component D of $\Delta {\setminus } S$ does not contain a lattice point in its interior and does not intersect $\partial \Delta$. In this 2-to-1 case, each component ${\tilde{D}}_1, {\tilde{D}}_2$ of $C {\setminus } {\mathcal {N}}$ is mapped homeomorphically by p onto the component of $\Delta ^\circ {\setminus } S$ corresponding to D. Since D is assumed to contain no lattice points in the interior, it follows that the corresponding component of $\Delta ^\circ {\setminus } S$ is simply-connected, and hence ${\tilde{D}}_1, {\tilde{D}}_2$ are as well.

Lemma 11.3.1 implies that no component of $\Delta {\setminus } S$ contains an interior lattice point, and so it remains only to be seen that every component of $C {\setminus } {\mathcal {N}}$ that corresponds to a component of $\Delta {\setminus } S$ intersecting $\partial \Delta$ is simply-connected. Let ${\tilde{D}}\subset C{\setminus } {\mathcal {N}}$ be such a component, and let D be the corresponding component of $\Delta ^\circ {\setminus } S$. Observe that ${\tilde{D}}$ is constructed by attaching two copies of D along $D \cap \partial \Delta$. It follows that ${\tilde{D}}$ is simply-connected if and only if $D \cap \partial \Delta$ is connected. The result now follows by Lemma 11.3.2. $\square$

Applicability of Theorem 9.5 It remains to verify the properties (1)–(4) of Theorem 9.5. By Theorem 10.4, for an A-curve A(v), the associated Dehn twist $T_{A(v)} \in \Gamma$. By Theorem 10.5, any curve $B(\xi ) \in {\mathcal {N}}$ arising from a primitive integer segment $\xi$ also satisfies $T_{B(\xi )} \in \Gamma$. It follows from the definitions that any curve c in any connected network is necessarily non-separating. For a nonseparating curve $c \subset C$, the Dehn twist $T_c \in \Gamma$ only if the associated spin structure satisfies $\phi (c) = 0$. Hence (1) holds.

For (2), we take a $\kappa$-standard embedding of $\Delta$. It is now easy to find a collection of curves ${\mathscr {S}}_{2r +4}$ determining the configuration ${\mathscr {D}}_{2 r + 3} \cup \{\Delta _1\}$ of Corollary 2.7. We take $a= B((0,0),(0,-1))$ and $a' = B((0,0),(0,1))$. Since $\Delta _a$ is assumed to be r-divisible, the edge of $\Delta _a$ lying along the x-axis extends at least as far as (r, 0). For $1 \le k \le r+1$, we can therefore take $c_{2k-1}$ to be $A(0,k-1)$, and for $1 \le k \le r$, we take $c_{2k}$ to be $B((k-1,0),(k,0))$. We take $a_{r+1}= B(\tau ) = B((r,0),(0,-1))$. The segments connecting $(0,-1),(0,0), (1,0), \dots , (d,0), (0,-1)$ separate $\Delta$ into two components, hence under the inflation procedure, the associated B-curves separate C. From the construction it is clear that the curves bound a subsurface of genus 0 with $r+2$ boundary components, as required for the configuration ${\mathscr {D}}_{2r+3} \cup \{\Delta _1\}$ of Corollary 2.7.

For (3), we observe that from the construction, the $\Delta _0$ curve of the configuration ${\mathscr {D}}_{2r+3}$ corresponds to $b\,{:}{=}\, B((-1,1),(0,1))$ on $\Delta$. One sees that A(0, 1) intersects this curve, and is an element of ${\mathcal {N}}$ as needed.

For (4), we begin by observing that only the element $A(0,1) \in {\mathcal {N}}$ intersects $B((-1,1),(0,1))$. Enumerate the components of $C {\setminus } {\mathcal {N}}$ as $\{D_i\}$. We claim that there are exactly three disks $D_1, D_2, D_3$ in $C {\setminus } {\mathcal {N}}$ with boundary lying on A(0, 1), and that $b \subset \overline{D_1} \subset C$. Indeed, using the notation of item (2) in the definition of ${\mathcal {N}}$, the disks $D_2$ and $D_3$ arise via inflation from the component of $\Delta {\setminus } S$ bounded by the triangle formed by $e', \sigma$, and the primitive integer segment(s) connecting $\kappa$ to w. Neither $D_2$ nor $D_3$ intersects b, and the only curve in ${\mathcal {N}}$ intersecting b is A(0, 1); this implies that $b \subset \overline{D_1}$ as claimed.

Thus, in $(C {\setminus } {\mathcal {N}}) \cup A(0,1)$, the disks $D_2$ and $D_3$ are joined into a single disk $D^+$, while $D_1$ has two portions of its boundary joined to create an annulus with core curve b. Upon passing to $(C{\setminus } \{b\}) {\setminus } {\mathcal {N}}'$, this annulus is cut open to create two annular regions bounded by b, while the disks $D^+$ and $D_i$ for $i \ge 4$ are unaffected. Thus ${\mathcal {N}}'$ does determine a filling network on $C {\setminus } \{b\}$ as required. Arboreality of ${\mathcal {N}}'$ was established in Lemma 11.2.

From admissible twists to vanishing cycles In order to address Question 1.2, it is necessary to better understand the relationship between admissible twists and vanishing cycles. A first remark is that any vanishing cycle is necessarily an admissible curve, so it remains only to show the converse. We observe that if $\alpha$ is a loop in ${\mathcal {M}}({\mathcal {L}})$ based at $C_0$ that determines a vanishing cycle, then any conjugate $\beta \alpha \beta ^{-1}$ also determines a vanishing cycle. To complete the argument, it therefore suffices to establish the following claim.

Lemma 11.5

Let a be any admissible curve on $C_0$. Then $T_a$ is conjugate in $\Gamma$ to some twist $T_c$ for c a vanishing cycle.

Proof

An admissible curve a determines a vertex in the graph $\mathcal C_\phi ^1(C_0)$ of Sect. 7. Theorems 10.4 and 10.5 together imply that $\Gamma$ has a generating set consisting entirely of vanishing cycles. Thus the set of vertices in ${\mathcal {C}}_\phi ^1(C_0)$ corresponding to vanishing cycles is nonempty.

We claim that if $a \in {\mathcal {C}}_\phi ^1(C_0)$ is adjacent to some vanishing cycle c, then a is also a vanishing cycle. Indeed, the condition that a and c are adjacent in ${\mathcal {C}}_\phi ^1(C_0)$ is equivalent to $i(a,c) = 1$, and hence by the braid relation,

$$\begin{aligned} T_a = (T_c T_a) T_c (T_c T_a)^{-1}. \end{aligned}$$

As $T_c, T_a \in \Gamma$ by the first part of Theorem A, the above observation implies that $T_a$ is a vanishing cycle. The claim now follows from the connectivity of ${\mathcal {C}}_\phi ^1(C_0)$ established in Lemma 7.5.

This concludes the proof of Theorem A. $\square$