On the Non-vanishing of the Powers of the Euler Class for Mapping Class Groups

Abstract

The mapping class group of an orientable closed surface with one marked point can be identified, by the Nielsen action, with a subgroup of the group of orientation-preserving homeomorphisms of the circle. This inclusion pulls back the “discrete universal Euler class” producing a non-zero class in the second integral cohomology of the mapping class group. In this largely expository note, we determine the non-vanishing behavior of the powers of this class. Our argument relies on restricting the cohomology classes to torsion subgroups of the mapping class group.

1 Introduction

Let ${\Gamma }_g^k$ denote the pure mapping class group of a closed orientable surface $\Sigma _g$ of genus $g \ge 1$ with $k\ge 0$ marked points. Homological properties of the mapping class group of surfaces of finite type have been studied for the last 40 years. For instance, cohomology classes of mapping class groups correspond to characteristic classes of surface bundles. Furthermore, for surfaces of genus $g\ge 2$, the rational cohomology of $\Gamma _g^k$ coincides with the cohomology of the moduli space ${\mathcal {M}}_{g,k}$ of Riemann surfaces of genus g with k marked points.

Some of the first homological calculations for mapping class groups are due to Harer. He computed $H_2(\Gamma _g^k;{{\,\mathrm{{\mathbb {Z}}}\,}})$ for genus $g\ge 5$ in [9] and proved a remarkable homological stability theorem in [10], which was a key result in the proof of Mumford’s conjecture for $H^*({\mathcal {M}}_g;{\mathbb {Q}})$ by Madsen and Weiss [19]. Miller [21] and Morita [23] constructed non-trivial cohomology classes in $H^*(\Gamma _g^k;{\mathbb {Q}})$, while Glover and Mislin [6] used torsion subgroups of the mapping class groups to detect torsion in their cohomology. In the same spirit, we use torsion elements in the mapping class group $\Gamma _g^1$ of a surface of genus $g\ge 1$ with one marked point to show the non-vanishing of some classes in $H^*(\Gamma _g^1;{\mathbb {Z}})$.

For $g\ge 2$, Nielsen defined a faithful action of ${\Gamma }_g^1$ on the circle ${{\,\mathrm{{\mathbb {S}}^1}\,}}$ which identifies ${\Gamma }_g^1$ with a subgroup of the group ${{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}})$ of orientation-preserving homeomorphisms of the circle (see, for example, [8] and Sect.  4). This monomorphism $\rho : {\Gamma }_g^1\hookrightarrow {{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}})$ pulls back the discrete universal Euler class ${{\,\mathrm{{\mathbf {E}}}\,}}$ and its powers ${{\,\mathrm{{\mathbf {E}}}\,}}^n$ to ${\Gamma }_g^1$ producing classes $\rho ^*({{\,\mathrm{{\mathbf {E}}}\,}}^n)=:\text {E}^n\in H^{2n}(\Gamma _g^1; {\mathbb {Z}})$ for each $n\ge 1$. As we review in Sect. 2, the nth cup product powers ${{\,\mathrm{{\mathbf {E}}}\,}}^n$ are known to be non-trivial torsion free cohomology classes in $H^{2n}({{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}});{{\,\mathrm{{\mathbb {Z}}}\,}})$, for $n\ge 1$. In this note, we determine the non-vanishing behavior of the powers $\text {E}^n\in H^{2n}({\Gamma }_g^1;{\mathbb {Z}})$ of the Euler class $\text {E}$ for ${\Gamma }_g^1$.

Theorem A

For $g\ge 1$ and $n\ge 1$, the cohomology classes $\text {E}^n\in H^{2n}({{\Gamma _g^1}; {\mathbb {Z}}})$ are nonzero. Furthermore, when $n \ge g$, the subgroup of $H^{2n}({\Gamma _g^1; {\mathbb {Z}}})$ generated by the class $\text {E}^n$ is a finite cyclic group of order a multiple of $4g(2g+1)$.

For genus $g=1$, Theorem A holds and more is known: the powers of the Euler class for $\Gamma _1^1\cong {{\,\mathrm{SL}\,}}(2,{{\,\mathrm{{\mathbb {Z}}}\,}})$ behave like the pull back of ${{\,\mathrm{{\mathbf {E}}}\,}}^n$ to a finite cyclic subgroup of ${{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}})$ (see Proposition 3.1). The group ${{\,\mathrm{SL}\,}}(2,{{\,\mathrm{{\mathbb {Z}}}\,}})$ acts faithfully on rays starting at the origin in the Euclidean plane. The corresponding monomorphism $\rho :{{\,\mathrm{SL}\,}}(2,{{\,\mathrm{{\mathbb {Z}}}\,}})\hookrightarrow {{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}})$ pulls back the class ${{\,\mathrm{{\mathbf {E}}}\,}}$ to a generator $x:=\rho ^*({{\,\mathrm{{\mathbf {E}}}\,}})$ in $H^2({{\,\mathrm{SL}\,}}(2,{\mathbb {Z}});{\mathbb {Z}})$ of the cohomology ring

$$\begin{aligned} H^*({{\,\mathrm{SL}\,}}(2,{\mathbb {Z}});{\mathbb {Z}})\cong {\mathbb {Z}}[x]/ \langle 12x\rangle . \end{aligned}$$

Therefore, the Euler class for ${{\,\mathrm{SL}\,}}(2,{{\,\mathrm{{\mathbb {Z}}}\,}})$ and all its powers are non-trivial torsion classes of order 12. In contrast, for genus $g\ge 3$ the Euler class $\text {E}\in H^2({\Gamma }^1_g;{\mathbb {Z}})$ is known to be non-trivial of infinite order. This also follows from our Theorem A and the universal coefficient theorem since the group $\Gamma _g^1$ is perfect.

We will observe, in Sects. 3 and 4, that the non-triviality of the classes $\text {E}^n$ is obtained by restricting the cohomology classes to a torsion subgroup of ${\Gamma }_g^1$ where the corresponding cohomology classes are known to be non-trivial. Our torsion bound in Theorem A comes from the order of specific torsion elements in $\Gamma _g^1$. We end this note with Sect. 5 where we comment on known related results. In particular, Theorems 5.3 and 5.4 give a partial understanding of the behavior of the powers of the “Euler classes” for the pure mapping class groups with one or more marked points.

Further work. In work in progress, we use the non-vanishing result Theorem A as a starting point to address the more subtle problem of computing the order of the classes $\text {E}^n$ for $n\ge g$ and showing that for $n<g$ the powers $\text {E}^n$ are torsion free cohomology classes. In [13], the order of $\text {E}^n$ at the threshold dimension $n=g$ was computed by the first author to be $2g(2g+1)$. The difference with the bound obtained in Theorem A is explained in our current work. It arises due to the fact that the computation in [13] was obtained not for the Nielsen action, but for an action that we refer to as the “projective action”.

We briefly and informally describe our current research. The approach is similar but more directly geometric than that of [13], and the aim is to provide an elementary, and unified description of the behavior of the powers of the Euler class for the pure mapping class groups with one or more marked points. The group $\Gamma _g^1$ acts naturally on the infinite simplex with vertex points of the circle. The action of $\Gamma _g^1$ on the circle gives rise to a bi-simplicial set which realizes $B\Gamma _g^1,$ and an associated double-chain complex which computes its homology. That is the setting for our computations. The elements S and T, described in Sect. 4, allow us to construct in the bi-complex, a 2n-chain, dual to $\text {E}^n,$ for $n \le g$. We seek to determine how this 2n-chain transitions from its behavior for $n<g$, where it extends to a cycle (which we know to be non-trivial by Theorem A), to one at the threshold $n=g$ where that fails.

2 The Universal Euler Class and Its Powers

Consider the group ${{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}})$ of orientation-preserving homeomorphisms and the group of lifts $\widetilde{{{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}})}$ with respect to the universal cover $\pi :{\mathbb {R}}\rightarrow {{\,\mathrm{{\mathbb {S}}^1}\,}}$. There is an epimorphism $p: \widetilde{{{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}})}\rightarrow {{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}})$ with kernel isomorphic to ${{\,\mathrm{{\mathbb {Z}}}\,}}$ generated by $T:{\mathbb {R}}\rightarrow {\mathbb {R}}$ the integral translation $T(x)=x+1$. This defines a non-split central extension of ${{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}})$

$$\begin{aligned} 0\rightarrow {{\,\mathrm{{\mathbb {Z}}}\,}}\rightarrow \widetilde{{{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}})}\xrightarrow {p} {{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}})\rightarrow 1, \end{aligned}$$

(1)

which is universal, in the sense of [22, Sect. 5], with kernel isomorphic to the Schur multiplier $H_2({{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}});{{\,\mathrm{{\mathbb {Z}}}\,}})$. The central extension (1) corresponds to a non-trivial generator ${{\,\mathrm{{\mathbf {E}}}\,}}$ in $H^2({{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}});{\mathbb {Z}})\cong {{\,\mathrm{{\mathbb {Z}}}\,}}$ that we refer as the discrete universal Euler class.

To better understand the powers ${{\,\mathrm{{\mathbf {E}}}\,}}^n$ of the discrete universal Euler class, let us recall the following classical result due to Mather [20] and Thurston [24]. Given M an orientable manifold, we distinguish the group of orientation-preserving homeomorphisms of M with the discrete topology ${{\,\mathrm{Homeo_+}\,}}(M)_\delta$ from the topological group ${{\,\mathrm{Homeo_+}\,}}(M)_\tau$ with the compact-open topology.

Theorem 2.1

(Thurston–Mather) Let M be any orientable manifold. The identity map $\text {id}:{{\,\mathrm{Homeo_+}\,}}(M)_\delta \rightarrow {{\,\mathrm{Homeo_+}\,}}(M)_\tau$ induces a continuous function between classifying spaces

$$\begin{aligned} \eta : {{\,\mathrm{BHomeo}\,}}_+(M)_{\delta }\rightarrow {{\,\mathrm{BHomeo}\,}}_+(M)_{\tau } \end{aligned}$$

which is a homology equivalence.

The subgroup of rotations $SO(2,{\mathbb {R}})$ is canonically identified with ${{\,\mathrm{{\mathbb {S}}^1}\,}}$ and the inclusion ${{\,\mathrm{{\mathbb {S}}^1}\,}}\hookrightarrow \text {Homeo}_+({{\,\mathrm{{\mathbb {S}}^1}\,}})_{\tau }$ is a homotopy equivalence; see, for example, [5, Prop. 4.2]. Then $\text {BHomeo}_+({{\,\mathrm{{\mathbb {S}}^1}\,}})_{\tau } \simeq \text {B}{{\,\mathrm{{\mathbb {S}}^1}\,}}\simeq \mathbb {CP}^\infty$ and its cohomology ring is given by

$$\begin{aligned} H^*(\text {BHomeo}_+({{\,\mathrm{{\mathbb {S}}^1}\,}})_{\tau };{{\,\mathrm{{\mathbb {Z}}}\,}})\cong {{\,\mathrm{{\mathbb {Z}}}\,}}[x], \end{aligned}$$

where x is a generator of $H^2(\text {BHomeo}_+({{\,\mathrm{{\mathbb {S}}^1}\,}})_{\tau };{{\,\mathrm{{\mathbb {Z}}}\,}})\cong {{\,\mathrm{{\mathbb {Z}}}\,}}$. Hence, for the circle ${{\,\mathrm{{\mathbb {S}}^1}\,}}$, the Thurston–Mather theorem and the universal coefficient theorem imply the following result.

Corollary 2.2

The cohomology ring of the discrete group ${{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}})$ is a polynomial ring generated by the universal Euler class ${{\,\mathrm{{\mathbf {E}}}\,}}\in H^2({{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}});{\mathbb {Z}})$, i.e.

$$\begin{aligned} H^*({{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}});{\mathbb {Z}})=H^*({{\,\mathrm{BHomeo}\,}}_+({{\,\mathrm{{\mathbb {S}}^1}\,}})_\delta ;{\mathbb {Z}})\cong {\mathbb {Z}}[{{\,\mathrm{{\mathbf {E}}}\,}}]. \end{aligned}$$

Therefore, all the powers ${{\,\mathrm{{\mathbf {E}}}\,}}^n$ are non-trivial torsion-free cohomology classes. See also Sect. 5 in [13] for a different proof.

3 Torsion and Non-triviality of the Powers of the Euler Class

We observe next that, when restricted to a finite cyclic subgroup ${{\,\mathrm{{\mathbb {Z}}}\,}}/k{{\,\mathrm{{\mathbb {Z}}}\,}}$, the powers of the universal discrete Euler class pull back to non-trivial torsion classes. Recall that the cohomology ring of ${{\,\mathrm{{\mathbb {Z}}}\,}}/k{{\,\mathrm{{\mathbb {Z}}}\,}}$ is known to be

$$\begin{aligned} H^*({{\,\mathrm{{\mathbb {Z}}}\,}}/k{{\,\mathrm{{\mathbb {Z}}}\,}};{{\,\mathrm{{\mathbb {Z}}}\,}})\cong {{\,\mathrm{{\mathbb {Z}}}\,}}[x]/\langle kx\rangle , \text { where }x\text { is a generator of } H^2({{\,\mathrm{{\mathbb {Z}}}\,}}/k{{\,\mathrm{{\mathbb {Z}}}\,}};{{\,\mathrm{{\mathbb {Z}}}\,}})\cong {{\,\mathrm{{\mathbb {Z}}}\,}}/k{{\,\mathrm{{\mathbb {Z}}}\,}}. \end{aligned}$$

Proposition 3.1

For any monomorphism $\phi : {\mathbb {Z}}/k{\mathbb {Z}}\hookrightarrow {{\,\mathrm{Homeo_+}\,}}{{\,\mathrm{{\mathbb {S}}^1}\,}}$, the pull-back of the discrete universal Euler class $\phi ^*({{\,\mathrm{{\mathbf {E}}}\,}})\in H^2({\mathbb {Z}}/k{\mathbb {Z}};{{\,\mathrm{{\mathbb {Z}}}\,}})$ is a generator of the cohomology ring $H^*({\mathbb {Z}}/k{\mathbb {Z}};{{\,\mathrm{{\mathbb {Z}}}\,}})$. In particular, all the powers $(\phi ^*({{\,\mathrm{{\mathbf {E}}}\,}}))^n=\phi ^*({{\,\mathrm{{\mathbf {E}}}\,}}^n)$ are non-trivial torsion classes in $H^{2n}({\mathbb {Z}}/k{\mathbb {Z}};{{\,\mathrm{{\mathbb {Z}}}\,}})$ of order k.

Proof

Consider first the finite cyclic group ${\mathbb {Z}}/k{\mathbb {Z}}$ acting faithfully on the circle by a rotation of angle $2\pi /k$. Then the pull back of the universal central extension (1) of ${{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}})$ is the non-split central extension

$$\begin{aligned} 0\rightarrow {\mathbb {Z}}\xrightarrow {\times k} {{\,\mathrm{{\mathbb {Z}}}\,}}\rightarrow {{\,\mathrm{{\mathbb {Z}}}\,}}/k{{\,\mathrm{{\mathbb {Z}}}\,}}\rightarrow 1 \end{aligned}$$

of ${{\,\mathrm{{\mathbb {Z}}}\,}}/k{{\,\mathrm{{\mathbb {Z}}}\,}}$ by ${{\,\mathrm{{\mathbb {Z}}}\,}}$ which corresponds to $\phi ^*({{\,\mathrm{{\mathbf {E}}}\,}})$ and is a generator of $H^2({\mathbb {Z}}/k{\mathbb {Z}};{{\,\mathrm{{\mathbb {Z}}}\,}})\cong {{\,\mathrm{{\mathbb {Z}}}\,}}/k{{\,\mathrm{{\mathbb {Z}}}\,}}$; see for example, [18, Ch. IV.7]. The pull back of the powers ${{\,\mathrm{{\mathbf {E}}}\,}}^n$ correspond to generators of $H^{2n}({\mathbb {Z}}/k{\mathbb {Z}};{{\,\mathrm{{\mathbb {Z}}}\,}})\cong {{\mathbb {Z}}}/k{{\,\mathrm{{\mathbb {Z}}}\,}}$ by functoriality of the cup product.

Up to conjugacy, the rotation group $SO(2,{\mathbb {R}})$ is the only maximal compact subgroup of ${{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}})$; see, for instance, [5, Prop. 4.1]. As a consequence, any finite subgroup of ${{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}})$ is conjugate to a cyclic group of rotations, and the proposition is then true for any monomorphism $\phi : {\mathbb {Z}}/k{\mathbb {Z}}\hookrightarrow {{\,\mathrm{Homeo_+}\,}}{{\,\mathrm{{\mathbb {S}}^1}\,}}$. $\square$

Corollary 3.2

Let $\Gamma$ be a discrete group acting faithfully on ${{\,\mathrm{{\mathbb {S}}^1}\,}}$ by orientation-preserving homeomorphisms. Suppose that $\Gamma$ has a torsion element of order k.

• (a) For all $n\ge 1$, the pull back of ${{\,\mathrm{{\mathbf {E}}}\,}}^n$ is a non-trivial class in $H^{2n}(\Gamma ;{{\,\mathrm{{\mathbb {Z}}}\,}})$.

• (b) If the pull back of ${{\,\mathrm{{\mathbf {E}}}\,}}^n$ is a torsion class, it must have order a multiple of k.

Proof

By hypothesis, we have monomorphisms

By Proposition 3.1, for any $n\ge 1$ the pull back

$$\begin{aligned} (\psi \circ \iota )^*({{\,\mathrm{{\mathbf {E}}}\,}}^n)=\iota ^*\left( \psi ^*({{\,\mathrm{{\mathbf {E}}}\,}}^n)\right) =\iota ^*\left( (\psi ^*({{\,\mathrm{{\mathbf {E}}}\,}}))^n\right) \end{aligned}$$

is a non-trivial class in $H^{2n}({\mathbb {Z}}/k{\mathbb {Z}};{{\,\mathrm{{\mathbb {Z}}}\,}})$ of order k. Therefore, the class $(\psi ^*({{\,\mathrm{{\mathbf {E}}}\,}}))^n$ is a non-trivial class in $H^{2n}(\Gamma ;{{\,\mathrm{{\mathbb {Z}}}\,}})$. If $(\psi ^*({{\,\mathrm{{\mathbf {E}}}\,}}))^n$ is a torsion class, then it must have order a multiple of k since $\iota ^*$ takes it to a torsion class of order k. $\square$

4 Powers of the Euler Class for Mapping Class Groups

Let $\Sigma _g$ denote the closed orientable surface of genus $g \ge 1$ and $z\in \Sigma _g$. The mapping class group ${\Gamma }_g^1$ with one marked point is the group of orientation-preserving homeomorphisms of $\Sigma _g$ modulo isotopy, where the point z is required to stay fixed under isotopies.

Consider the presentation of the fundamental group

$$\begin{aligned} \pi _1(\Sigma _g, z)=\langle a_1, a_2,\ldots ,a_{2g}\vert \ a_1 \cdots a_{2g} \cdot {a_1^{-1}} \cdots {a_{2g}^{-1}} = 1\rangle . \end{aligned}$$

The Dehn–Nielsen–Baer theorem identifies ${\Gamma }_g^1$ with an index 2 subgroup of the automorphism group ${{\,\mathrm{Aut}\,}}\left( \pi _1(\Sigma _g, z)\right)$ (see, for example, [4, Ch. 8]). Under this identification, the automorphisms of $\pi _1(\Sigma _g, z)$

represent torsion elements of ${\Gamma }_g^1$ or order 4g and $2g+1$, respectively. Geometrically they are related to the 4g-gon and $2(2g+1)$-gon symmetries of the surface $\Sigma _g$.

For $g=1$, the Dehn–Nielsen–Baer theorem implies $\Gamma _1^1\cong {{\,\mathrm{SL}\,}}(2,{{\,\mathrm{{\mathbb {Z}}}\,}})$ and we have that

$$\begin{aligned} S= \begin{pmatrix} 0 &{}\quad -1 \\ 1 &{}\quad 0 \end{pmatrix} \text { and }T= \begin{pmatrix} 0 &{}\quad -1 \\ 1 &{}\quad -1 \end{pmatrix} \end{aligned}$$

of orders 4 and 3, respectively.

For $g\ge 2$, Nielsen defined a faithful action of ${\Gamma }_g^1$ on ${{\,\mathrm{{\mathbb {S}}^1}\,}}$ by orientation-preserving homeomorphisms that we briefly describe next; see [8] and [4] for a more detailed discussion. Fixing a hyperbolic metric on $\Sigma _g$, its universal cover can be identified with the hyperbolic disk ${\mathbb {D}}^2$, which has a natural compactification to a closed disc. Let ${\tilde{z}}\in {\mathbb {D}}^2$ be a distinguished lift of the marked point $z\in \Sigma _g$. For $f\in {{\,\mathrm{Homeo_+}\,}}(\Sigma _g)$ fixing the marked point z let ${\tilde{f}}$ denote the unique lift of f to ${\mathbb {D}}^2$ that fixes ${\tilde{z}}$. It can be shown that the action of ${\tilde{f}}$ on ${\mathbb {D}}^2$ extends to a homeomorphism of the boundary circle, which depends only on the isotopy class of f. This procedure gives a well-defined monomorphism

$$\begin{aligned} \rho : {\Gamma }_g^1 \hookrightarrow {{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}}), \end{aligned}$$

which is the Nielsen action of ${\Gamma }_g^1$ on ${{\,\mathrm{{\mathbb {S}}^1}\,}}$.

For genus $g \ge 2$, the Gromov boundary of $\pi _1(\Sigma _g,z)$ is a topological circle ${{\,\mathrm{{\mathbb {S}}^1}\,}}$, on which the group of automorphisms ${{\,\mathrm{Aut}\,}}(\pi _1(\Sigma _g,z))$ acts faithfully by homeomorphisms. With the Dehn–Nielsen–Baer identification, this boundary action is conjugate to the geometric action that we just described.

The Euler class for ${\Gamma }_g^1$ is defined as $\text {E}:=\rho ^*({{\,\mathrm{{\mathbf {E}}}\,}})\in H^2({\Gamma }_g^1;{{\,\mathrm{{\mathbb {Z}}}\,}})$, the pull back of the discrete universal Euler class under the standard action. This cohomology class corresponds to the central extension

$$\begin{aligned} 1\rightarrow {\mathbb {Z}}\rightarrow \Gamma _{g,1}\rightarrow {\Gamma }_g^1\rightarrow 1, \end{aligned}$$

where $\Gamma _{g,1}$ denotes the mapping class group of an orientable surface $\Sigma _{g,1}$ of genus g with one boundary component (find more details in [4, Ch 5.5] and [7]). The epimorphism above is induced from the inclusion $\Sigma _{g,1}\simeq \Sigma _{g}-N_{\epsilon }(z)\hookrightarrow \Sigma _g^1$, where $N_{\epsilon }(z)=\{x\in \Sigma _{g}^1: d(x,z)<\epsilon \}$ for a small $\epsilon >0$. The kernel is generated by a Dehn twist around the boundary component, which is the simple loop $\partial N_{\epsilon }(z)$ around the marked point z.

Proof of Theorem A

The mapping classes S and T generate torsion cyclic subgroups in ${\Gamma }_g^1$ of order $k=4g$ and $k=2g+1$, respectively. For $g=1$, the group ${\Gamma }_1^1\cong {{\,\mathrm{SL}\,}}(2,{\mathbb {Z}})$ acts faithfully on rays starting at the origin in the Euclidean plane, and for genus $g\ge 2$ the Nielsen action is faithful. It follows from Corollary 3.2 a) that the powers $\text {E}^n$ are non-trivial for all $n\ge 1$. For $n\ge g$, the powers $\text {E}^n$ are known to vanish over the rationals (Theorem 5.3); hence, the classes $\text {E}^n$ are torsion and by Corollary 3.2 b) must have order a common multiple of $k=4g$ and $k=2g+1$. $\square$

Remark

We can apply Corollary 3.2 a) to any finite cyclic subgroup of $\Gamma _g^1$. The order of a finite cyclic subgroup of ${\Gamma }_g^1$ is known to be at most $4g+2$ and this upper bound is attained for $g\ge 2$ (see, for example [4, Cor. 7.6]). Since 4g and $2g+1$ are relatively prime, the subgroups generated by S and T that we consider in the proof of Theorem A give us the largest lower bound $4g(2g+1)$ for torsion that we could find with these elements.

If other torsion elements of $\Gamma _g^1$ are known, one can try to look for a better lower bound. For instance, if $g=p-1$ with p a prime number, then $\Gamma _g^1$ has p-torsion [16, Thm 2.7] and from Corollary 3.2 b), we obtain that the cohomology classes $\text {E}^{n}$ have order a common multiple of 4g, $2g+1$ and $g+1$, when $n\ge g$.

5 Related Results

For $g \ge 0$ and $k\ge 1$, consider $z_1, z_2,...,z_{k} \in \Sigma _g$. The pure mapping class group ${\Gamma }_g^ k$ of $\Sigma _g$ is the group of orientation-preserving homeomorphisms of $\Sigma _g$, modulo isotopy, where the points $z_i$ are required to stay fixed under isotopies. For $g\ge 2$ and $k\ge 0$, the moduli space ${\mathcal {M}}_{g,k}$ of genus g Riemann surfaces with k marked points is a rational model for the classifying space ${B}\Gamma _{g}^k$; therefore, $H^*({\mathcal {M}}_{g,k};{\mathbb {Q}})\cong H^*(\Gamma _{g}^k;{\mathbb {Q}})$. For $k=0$, we use the notation $\Gamma _g$ and ${\mathcal {M}}_g$. Research in the last 40 years has been motivated by the following general problem:

Problem 5.1

Compute the groups $H^i(\Gamma _g^k;{\mathbb {K}})$ for all g, k and i and understand the ring structure of $H^*(\Gamma _g^k;{\mathbb {K}})$, for coefficients ${\mathbb {K}}={\mathbb {Q}}$ and ${{\,\mathrm{{\mathbb {Z}}}\,}}$.

We comment briefly on some of the most remarkable results towards the answer of Problem 5.1. Harer proved in [10] that the cohomology groups $H^i(\Gamma _g^k;{\mathbb {Z}})$ are independent of the genus g and k in degrees small relative to i. The range where this happens is called the “stable range” and has been improved over the years. Mumford conjectured, and Madsen and Weiss proved in [19], that in low cohomological degrees $H^*({\mathcal {M}}_g;{\mathbb {Q}})$ is a polynomial algebra in classes $\kappa _i$ of degree 2i, giving a complete picture of the cohomology ring $H^*({\mathcal {M}}_g;{\mathbb {Q}})$ in the stable range. Outside of the stable range, a few of the cohomology groups are known, even rationally. For instance, there are complete computations for genus $g\le 4$ and for low cohomology degrees; see, for example, [1] for computations of the cohomology ring of $\Gamma _2$.

The study of the cohomology groups and their ring structure in the “unstable range” is an active area of research. The $\kappa$-classes mentioned before are examples of tautological classes of ${\mathcal {M}}_{g,k}$, cohomology classes “naturally coming from geometry”. An important direction of research is given by the Faber conjectures [3] which describe the structure of the ring generated by the tautological classes.

The “Euler classes” for the pure mapping class group. For $k\ge 1$, once the marked points $z_1,...,z_{k}$ are fixed in $\Sigma _g$ we will distinguish the $\Gamma _g^1$s by writing $\Gamma _g^{z_i}$ for the mapping class group with the single marked point $z_i.$ For $1\le i\le k$, each $\Gamma _g^{z_i}$ is a distinct quotient group of ${\Gamma }_g^ k$: two elements of ${\Gamma }_g^ k$ determine the same element of $\Gamma _g^{z_i}$ if they are isotopic by an isotopy which keeps $z_i$ fixed. Let $p_i: {\Gamma }_g^ k \rightarrow \Gamma _g^{z_i}$ denote the quotient homomorphism. As already mentioned in Sect. 4, the $\Gamma _g^{z_i}$ can be identified with a subgroup of ${{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}})$. The composite

$$\begin{aligned} \rho _i: \Gamma _g^{k} \xrightarrow {p_i} \Gamma _g^{z_i} \hookrightarrow {{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}}) \end{aligned}$$

(2)

pulls back the powers ${{\,\mathrm{{\mathbf {E}}}\,}}^n$ producing the classes $\text {E}_i^n \in H^{2n}({\Gamma }_g^ k;{\mathbb {Z}})$ for all $1\le i\le k$ and $n\ge 1$. In trying to contribute to a partial answer to Problem 5.1, our work focuses on the following problem:

Problem 5.2

Understand the behavior of the “Euler classes” $\text {E}_i$ and their powers $\text {E}_i^n\in H^{2n}(\Gamma _g^k;{{\,\mathrm{{\mathbb {Z}}}\,}})$ for $k\ge 1$ and $n\ge 1$.

Our Theorem A and Theorems 5.3 and 5.4 below partially answer this problem.

Harer computed in [9] the second integral homology group of $\Gamma _{g}^k$ for $g\ge 5$. It is known that $H_2(\Gamma _{g,r}^k;{\mathbb {Z}})\cong H_2(\Gamma _{g,r+k};{\mathbb {Z}})\oplus {\mathbb {Z}}^{k}$ for $g\ge 3$ and $r,k\ge 0$ [14, Prop. 1.4]. The summand ${\mathbb {Z}}^{k}$ of $H^2(\Gamma _{g}^k;{\mathbb {Z}})$ is generated by the cohomology classes $\text {E}_1,\text {E}_2\ldots ,\text {E}_k$.

For $k\ge 1$, the class $\text {E}_i \in H^2(\Gamma _g^k; {\mathbb {Q}})$ corresponds to the restriction of the $\psi _i$-class to ${{\mathcal {M}}}_{g,k}$ from its Deligne–Mumford compactification $\overline{{\mathcal {M}}}_{g,k}.$ The class $\psi _i$ is the first Chern class of the cotangent bundle bundle over $\overline{{\mathcal {M}}}_{g,k}$ associated with the marked point $z_i$; see for example [7]. The $\psi$-classes are also tautological classes of ${\mathcal {M}}_{g,k}$.

From the algebro-geometric perspective, a result by Ionel [11, Thm 0.1] establishes that any monomial in the tautological classes of degree at least g vanishes when restricted to $H^*({\mathcal {M}}_{g,k};{\mathbb {Q}})$ when $g \ge 2$ and $k\ge 1$. As a particular case, we have the following vanishing result over the rationals (which also follows from the work of Looijenga [15] for $k=1$).

Theorem 5.3

(Vanishing over ${\mathbb {Q}}$  [11]) For any $1\le i\le k$, the powers $\text {E}_i^n\in H^{2n}(\Gamma _g^k;{\mathbb {Q}})$ vanish for $n\ge g$.

Vanishing results in [12] and [13] for $n\ge g$ apply to the nth power of the Euler class of $\Gamma _g^1$ considered as an element of ${{\,\mathrm{Hom}\,}}(H_{2n}(\Gamma _g^1;{{\,\mathrm{{\mathbb {Z}}}\,}});{{\,\mathrm{{\mathbb {Z}}}\,}})$ under the universal coefficient theorem, agreeing with the results in Theorem  5.3 for $k=1$.

For genus $g\ge 2$ and $k\ge 1$, from Morita’s result [23, Thm 7.5], it follows that $\text {E}_1^n, \text {E}_2^n,\ldots \text {E}_k^n$ generate a summand of $H^{2n}(\Gamma _g^{k};{\mathbb {Q}})$ isomorphic to ${\mathbb {Q}}^k$ for $2n\le g/3$. This range has been improved to $2n\le g/2$ in [2, Cor 1.2]. Hence, there is a non-vanishing result for $g\ge 2$:

Theorem 5.4

(Non-vanishing and independence [2, 23]) For any $1\le i\le k$, the powers $\text {E}_i^n\in H^{2n}(\Gamma _g^{k};{\mathbb {Z}})$ are non-trivial torsion-free independent classes for $n\le g/4$.

For $k=1$, Theorem A improves on Theorem 5.4 by showing non-vanishing of $\text {E}^n$ for all $n\ge 1$. For $g\ge 1$, $k\ge 2$ and $1\le i\le k$ the composite $\rho _i$ described in (2) has torsion-free kernel and any torsion subgroup contained in $\Gamma _g^k$ injects in ${{\,\mathrm{Homeo_+}\,}}({{\,\mathrm{{\mathbb {S}}^1}\,}})$ through $\rho _i$. Hence, the strategy used in the proof of Theorem  A can be used to show non-vanishing of the classes $\text {E}_i^n$, as long as the pure mapping class group $\Gamma _g^k$ has non-trivial torsion subgroups. The lower bound for the order of $\text {E}_i^n$, when $n\ge g$, depends on the order of these torsion subgroups.

For genus $g=1, 2, 3$ and $k\ge 1$, Lu [17, Sect. 1] investigated the p-torsion in $\Gamma _g^k$, for p a prime number. From her computations, then non-vanishing of the classes $\text {E}_i^n$ and specific lower bounds for their order, when $n\ge g$ and $g=2,3$, can be obtained as long as $k< 2g+3$. On the other hand, from [16, Thm 2.7], the pure mapping class group $\Gamma _g^2$ contains p-torsion if and only if $\Gamma _g^1$ has p-torsion. Therefore, the cohomology classes $\text {E}_1^n, \text {E}_2^n\in H^{2n}(\Gamma _g^2;{{\,\mathrm{{\mathbb {Z}}}\,}})$ have order a multiple of $2g(4g+1)$.

In contrast, from [17, Lemma 1.1(i)] it follows that the group $\Gamma _{g}^{k}$ is torsion free for $k\ge 2g+3$. As a consequence, the torsion in cohomology disappears in high degree and the integral classes $\text {E}_i^n$ eventually vanish.