The fundamental group of a rigid Lagrangian cobordism

Abstract

In this article we extend the construction of the Floer fundamental group to the monotone Lagrangian setting (for weakly exact or monotone Lagrangians with large minimal Maslov number) and use it to study the fundamental group of a Lagrangian cobordism $W\subset (\mathbb {C}\times M, \omega _{st}\oplus \omega )$ between two Lagrangian submanifolds $L, L'\subset ( M, \omega )$. We show that under natural conditions the inclusions $L,L'\hookrightarrow W$ induce surjective maps $\pi _{1}(L)\twoheadrightarrow \pi _{1}(W)$, $\pi _{1}(L')\twoheadrightarrow \pi _{1}(W)$ and when the previous maps are injective then W is an h-cobordism. In particular, in dimension at least 6, W is topologically trivial in this case.

Résumé

Dans cet article, nous prolongeons la construction du groupe fondamental de Floer au cadre lagrangien monotone (faiblement exact ou monotone avec nombre de Maslov minimal suffisemment grand) et l’utilisons pour étudier le groupe fondamental d’un cobordisme lagrangien $W\subset (\mathbb {C}\times M, \omega _{st}\oplus \omega )$ entre deux sous-variétés lagrangiennes $L, L'\subset ( M, \omega )$. Nous montrons que dans des conditions naturelles les inclusions $L,L'\hookrightarrow W$ induisent des applications surjectives $\pi _{1}(L)\twoheadrightarrow \pi _{1}(W)$, $\pi _{1}(L')\twoheadrightarrow \pi _{1}(W)$ et quand les applications précédentes sont injectives W est un h-cobordisme. En particulier, en dimension au moins 6, W est topologiquement trivial dans ce cas.

1 Introduction

In [1] the first author recovered the fundamental group of a monotone symplectic manifold $(M, \omega )$ by studying the moduli spaces of “augmentations” (see, Definition 2.2). The first objective of this paper is to extend this construction to the Lagrangian setting: when no holomorphic disc can arise on the Lagrangian submanifold, the Lagrangian and Hamiltonian settings turn out to be very similar, and the same ideas allow to recover the fundamental group of a Lagrangian submanifold from Floer theoretical objects.

The second objective of this paper is to study the fundamental group of a Lagrangian cobordism, in particular a monotone or weakly exact cobordism with two ends.

Let $L, L' \subset (M, \omega )$ be two compact, connected Lagrangian submanifolds. Denote by $\tilde{M} = \mathbb {C}\times M$ the symplectic manifold with the two-form $\tilde{\omega }= dx\wedge dy \oplus \omega$ and denote by $\pi :\tilde{M}\rightarrow \mathbb {C}$ the projection.

Definition 1.1

A Lagrangian cobordism $(W; L, L')$ is a non-compact embedded Lagrangian $W\subset \tilde{M}$ such that for some $\epsilon > 0$ we have $\pi ^{-1}([\epsilon , 1-\epsilon ]\times \mathbb {R}) \cap W = \hat{W}$ is a smooth-compact cobordism between L and $L'$ and $W{\setminus } \hat{W}= (-\infty ,\epsilon )\times \{0\} \times L \bigcup (1-\epsilon , \infty ) \times \{0\} \times L'$.

The relation of Lagrangian cobordism between Lagrangian submanifolds was introduced by Arnold in [2, 3]. Immersed Lagrangian cobordism was studied in [4, 5] independently. Monotone embedded Lagrangian cobordism was studied in [6] and monotone embedded Lagrangian cobordism between more than two Lagrangians was studied in [7, 8]. The main examples of such cobordisms are given by the Lagrangian suspension construction [9] and by the trace of Lagrangian surgery [10,11,12,13].

1.1 Weakly exact and monotone Lagrangian

Let $L\subset M$ be a Lagrangian submanifold. There are two morphisms associated to L the symplectic area $\omega$ and the Maslov index $\mu$:

$$\begin{aligned} \omega : \pi _{2}(M, L) \rightarrow \mathbb {R},\quad \text { and }\quad \mu :\pi _{2}(M,L)\rightarrow \mathbb {Z}. \end{aligned}$$

The positive generator of the subgroup defined by the image of the homomorphism $\mu$ is called the minimal Maslov number of L and is denoted by $N_{L}$. We set $N_{L}=\infty$ when the subgroup is trivial. Lagrangians are assumed to be either

• weakly exact: $\omega \vert _{\pi _{2}(M,L)} = 0$ or

• monotone: there is a positive real number $\rho \in \mathbb {R}$ such that for every disc $u:(D^{2},S^{1})\rightarrow (M, L)$ we have $\omega (u)= \rho \mu (u)$.

We assume here that all monotone Lagrangians satisfy $N_{L}> \text {dim}(L)$+1.

When a Lagrangian cobordism $(W; L, L')$ is weakly exact or monotone, what we call rigid, its topology is very restricted. For example in [7, Theorem 2.2.2] the authors proved that a monotone Lagrangian cobordism $(W; L, L')$ where L and $L'$ are uniformly monotone¹ and $N_{L}, N_{L'} \ge 2$, is a quantum h-cobordism. This means that the Lagrangian quantum homology [14, Section 5],  [7, Section 5.1] satisfies $QH(W,L;\mathbb {Z}_{2})= 0 = QH(W,L';\mathbb {Z}_{2})$. The Lagrangian quantum homology is, roughly, the homology of a chain complex whose generators are the critical points of a Morse function and whose differential counts pearl trajectories, that are configurations of negative gradient flow lines and J-holomorphic discs, between the critical points (see, the Proof of Theorem 1.6 for more details on quantum homology). Moreover, in [7, Theorem 2.2.2] they proved that if $L, L'$ are wide² then the maps in singular homology $H_{1}(L;\mathbb {Z}_{2})\rightarrow H_{1}(W;\mathbb {Z}_{2})$ and $H_{1}(L';\mathbb {Z}_{2})\rightarrow H_{1}(W;\mathbb {Z}_{2})$ induced by the inclusions have the same image.

1.2 Main result

In this paper we investigate the maps induced by the inclusions $L,L'\hookrightarrow W$ on the fundamental groups, $\pi _{1}(L)\xrightarrow {i_{\sharp }} \pi _{1}(W)$ and $\pi _{1}(L')\xrightarrow {i_{\sharp }} \pi _{1}(W)$, and we prove the following:

Theorem 1.2

Let $(W;L,L')$ be a weakly exact Lagrangian cobordism or a monotone Lagrangian cobordism with $N_{W}> \text {dim} (W)$+1, then the inclusions $L, L'\hookrightarrow W$ induce surjective maps $\pi _{1}(L)\xrightarrow {i_{\sharp }} \pi _{1}(W)$ and $\pi _{1}(L')\xrightarrow {i_{\sharp }} \pi _{1}(W)$ on the fundamental groups.

Remark 1

One can expect these maps to be also injective, at least in many cases. Unfortunately, a better understanding of the relations in the fundamental group is still needed to answer this question.

Remark 2

Notice that we use the construction of the fundamental group to obtain these results, while a somewhat more “algebraic” proof was pointed out to us by Baptiste Chantraine [16], using local coefficients Floer homology. In a few words, if $i_{\sharp }$ is not onto, then in the universal cover $\tilde{W}$ of W, the preimage of L is not connected, which is detected by the fact that $H_{0}(L; \mathbb {Z}[\pi _{1}(W)])\ne \mathbb {Z}$, but for an exact Lagrangian, the second author proved in [17] that $H_{0}(L; \mathbb {Z}[\pi _{1}(W)])\cong H_{0}(W; \mathbb {Z}[\pi _{1}(W)])=\mathbb {Z}$. For weakly exact and monotone Lagrangian with $N_{W}> dim(W)+1$ this is still true and it follows from the Proof of Theorem 1.6).

As a consequence, the construction of the Floer fundamental group is not intrinsically required to prove the surjectivity of these maps. However, we believe that the construction of the fundamental group in both the Lagrangian and Lagrangian cobordism settings has some interest in itself, as well as this version of the proof: being slightly more geometric, this proof may also offer an interesting point of view to tackle the injectivity question.

A consequence of Theorem 1.2 is:

Corollary 1.3

If moreover $\pi _{2}(M,L)=0=\pi _{2}(M,L')$, then the maps $\pi _{1}(L)\xrightarrow {i_{\sharp }} \pi _{1}(W)$ and $\pi _{1}(L')\xrightarrow {i_{\sharp }} \pi _{1}(W)$ are isomorphisms.

In [17] the second author studied exact Lagrangian cobordism and she showed:

Theorem 1.4

[17, Theorem 18] Let $(W; L, L')$ be an exact, orientable and spin Lagrangian cobordism with $\mu \vert _{\pi _{2}(\tilde{M}, W)}=0$. If $L,L' \hookrightarrow W$ induce injective morphisms $\pi _{1}(L)\xrightarrow {i_{\sharp }} \pi _{1}(W)$ and $\pi _{1}(L')\xrightarrow {i_{\sharp }} \pi _{1}(W)$, then $(W; L, L')$ is an h-cobordism. Moreover, if $dim(W) \ge 6$ then there is a diffeomorphism $W\cong \mathbb {R}\times L$.

Recall that an h-cobordism is a cobordism $(W; L, L')$ where the maps defined by the inclusions $L, L' \hookrightarrow W$ are homotopy equivalences.

We have the following corollary to Theorem 1.4:

Corollary 1.5

Let $L,L'$ be Lagrangian submanifolds with $\pi _{2}(M,L)=0=\pi _{2}(M,L')$ and $(W;L,L')$ be an exact Lagrangian cobordism. If W is orientable and spin then it is an h-cobordism. Moreover, if $dim(W) \ge 6$ then there is a diffeomorphism $W\cong \mathbb {R}\times L$.

Remark 3

In particular, if $(W; L, L') \subset \tilde{M}$ where $M = T^{*}N$ is a cotangent bundle of a closed manifold N, then $\pi _{2}( T^{*}N, L) = 0 = \pi _{2}( T^{*}N, L')$, and then Corollary 1.5 applies. To see this, notice that by [18, 19] the map $\rho :L\hookrightarrow T^{*}N \rightarrow N$ is a homotopy equivalence, on the other hand the projection $T^{*}N\rightarrow N$ is a strong deformation retract, therefore the inclusion $L\hookrightarrow T^{*}N$ is a homotopy equivalence. In the same way one see $L'\hookrightarrow T^{*}N$ is a homotopy equivalence.

Using similar techniques to the ones in the proof of the first statement of Theorem 1.4, based on the Biran–Cornea machinery for Lagrangian cobordisms [7], we can show:

Theorem 1.6

Let $(W; L,L')$ be a weakly exact or a monotone (with $N_{W} > \text {dim}(W)+1$) orientable and spin Lagrangian cobordism. If $L,L' \hookrightarrow W$ induce injective morphisms $\pi _{1}(L)\xrightarrow {i_{\sharp }} \pi _{1}(W)$ and $\pi _{1}(L')\xrightarrow {i_{\sharp }}\pi _{1}(W)$, then W is an h-cobordism.

Remark 4

The previous theorem was proved in [17, Corollary 2] for exact Lagrangian cobordisms and it is based on a generalization of the result [7, Theorem 2.2.2]. In fact, we expect to prove that a weakly exact or a monotone (with $N_{W} > \text {dim}(W)+1$) spin Lagrangian cobordism $(W; L,L')$ with $\text {dim}(W)\ge 6$ is diffeomorphic to $\mathbb {R}\times L$. For general monotone Lagrangian cobordism this is not true, Haug [12, Theorem 1.5] constructed examples of monotone Lagrangian cobordisms with $N_{W} < \text {dim}(W)$ and that are not diffeomorphic to $\mathbb {R}\times L$.

The structure of the paper is as follows. In Sect. 2 we present the definition of the Floer fundamental group of a compact Lagrangian, adapting the construction in [1] to the Lagrangian setting. In Sect. 3 we adapt the Floer fundamental group to a non-compact Lagrangian with cylindrical ends. In Sect. 4 we give the proofs of Theorem 1.2 and Corollary 1.3 and Theorem 1.6.

2 Fundamental group of a Lagrangian submanifolds

2.1 Quick review of moduli spaces

Let $(M,\omega )$ be a tame symplectic manifold, this means that there is an almost complex structure J on M such that $g(\cdot , \cdot ) = \omega (\cdot , J\cdot )$ is a Riemannian metric and such that the Riemannian manifold (M, g) is complete, the sectional curvature of g is bounded and the injectivity radius bounded away from zero. The space of time dependent almost complex structures $J=(J_{t})_{t\in [0,1]}$ on M is denoted by $\mathcal {J}(M)$ (for genericity reasons, all the almost complex structures considered on the manifold M will be time dependent, but this dependency will be kept implicit to reduce notations).

Let $H: M\times [0, 1] \rightarrow \mathbb {R}$ be a Hamiltonian such that, if we denote by $(\phi ^{t})_{t\in [0,1]}$ the induced Hamiltonian isotopy, then $L_{1}=\phi ^{1}(L)$ satisfies $L\pitchfork L_{1}$.

An intersection point $x\in L\cap L_{1}$ is called fillable if the Hamiltonian trajectory that ends at x, considered as a path from L to L, bounds a disc $\sigma$ relative to L, i.e $[(\phi ^{t})^{-1}(x)]= 0\in \pi _{1}(M,L)$. Such fillings have a well defined Maslov index $\mu (\sigma )$.

Let $\mathcal {I}(L,L_{1})=\{x\in L\cap L_{1} \text { , }[(\phi ^{t})^{-1}(x)]= 0\in \pi _{1}(M,L)\}$ be the set of fillable intersection points. We denote by $\tilde{\mathcal {I}}(L,L_{1})$ the covering space with fiber $\pi _{2}(M,L)/\ker \mu$. An element $x= [(\underline{x},\sigma )]\in \tilde{\mathcal {I}}(L,L_{1})$ is an equivalence class of pairs $(\underline{x},\sigma )$ where $\underline{x}\in \mathcal {I}(L,L_{1})$ and $\sigma$ is a capping disc. The equivalence relation is given by $(\underline{x},\sigma )\sim (\underline{x}',\sigma ')$ if $\underline{x}=\underline{x}'$ and $\mu (\sigma )=\mu (\sigma ')$. We write

$$\begin{aligned} |x|=|(\underline{x},\sigma )| = \mu (\sigma ). \end{aligned}$$

Finally, we let $\tilde{\mathcal {I}}_{k}(L,L_{1})=\{x\in \tilde{\mathcal {I}}(L,L_{1}), |x|=k\}$.

Remark 5

Several versions of Floer trajectories can be used to define the differential of the Floer complex. The closest one to the point of view used by the first author in [1], is that of half tubes with boundary on L satisfying the non-homogeneous Floer equation. From this point of view, the absolute and relative cases are strictly parallel: the objects of the latter are all “halves” of the objects of the former with boundary in L and satisfy the same equations. All the arguments are the same, as long as no bubbling occurs on 0 and 1 dimensional moduli spaces.

However, the most convenient point of view for a cobordism oriented purpose like ours is that of strips satisfying “moving boundary conditions”. All the Floer trajectories, augmentations or co-augmentations are everywhere holomorphic and hence much easier to control in the cobordism framework. In consequence, the construction of the fundamental group in the relative case will be developed using this point of view. Aside the definition of the moduli spaces, this of course does not essentially affect the construction which remains parallel to the absolute case.

Let $J=(J_{t})_{t\in [0,1]}\in \mathcal {J}(M)$ with $J_{t}$ a $\omega$-compatible almost complex structure for each $t\in [0,1]$. We are interested in the Floer moduli spaces and their classical variants.

We fix once for all a smooth non-increasing function $\beta :\mathbb {R}\rightarrow \mathbb {R}$ such that

$$\begin{aligned} \beta (s)&=1\quad \text { for }s\le -1 \\ \beta (s)&=0\quad \text { for }s\ge 0 \end{aligned}$$

and use it to define several moving boundary conditions for maps $u \in C^{\infty }(\mathbb {R}\times [0,1], M)$ by considering the conditions:

for the following collection $\chi _{1},\dots ,\chi _{4}$ of cut-off functions derived from $\beta$ (see Fig. 1):

• 1. $\chi _{1}\equiv 1$, which defines the fixed boundary Floer equation,

• 2. $\chi _{2}(s)=\beta (s)$, which defines the augmentation equation,

• 3. $\chi _{3}(s)=\beta (-s)$, which defines the co-augmentation equation.

• 4. $\chi _{4,R}(s)=\beta (s-R)\beta (-s-R)$, which defines the R-perturbed strip equation, where $R\in [0,+\infty )$.

Fig. 1

Moving boundary conditions and associated moduli spaces

The energy of a map u is defined by:

$$\begin{aligned} E(u):= \iint \limits _{\mathbb {R}\times [0, 1]}\Vert \frac{\partial u}{\partial s}\Vert ^{2}dsdt. \end{aligned}$$

(1)

Finite energy solutions of ($F_{i}$) have converging ends, either to a point of L when $\chi _{i}(s)=0$ on this end or to an intersection point of L and $L_{1}$ when $\chi _{i}(s)=1$. Let $\mathcal {U}= \{u\in C^{\infty }(\mathbb {R}\times [0, 1], M) \text { }|\text { } E(u) < \infty \}$. Let $\star$ be a point in L and $x=[(\underline{x},\sigma _{x})], y=[(\underline{y},\sigma _{y})]\in \tilde{\mathcal {I}}(L,L_{1})$. We are interested in the moduli spaces below:

$$\begin{aligned} \mathcal {M}(y,x)&= \{u\in \mathcal {U}\quad |\quad (F_{1}), \lim _{s\rightarrow \pm \infty } u(s,t)=|{\begin{matrix}{\underline{x}}\\ {\underline{y}} \end{matrix}},[\sigma _{y}\sharp u\sharp {\bar{\sigma }}_{x}]=0\}\\ \mathcal {M}(x,\varnothing )&= \{u\in \mathcal {U}\quad |\quad (F_{2}), \lim _{s\rightarrow -\infty } u(s,t)={\underline{x}},[\sigma _{x}\sharp u]=0\}\\ \mathcal {M}(\star ,x)&= \{u\in \mathcal {U}\quad |\quad (F_{3}), \lim _{s\rightarrow \pm \infty } u(s,t)=|{\begin{matrix}{\underline{x}}\\ \star \end{matrix}},[u\sharp {\bar{\sigma }}_{x}]=0\}\\ \mathcal {M}(\star ,\varnothing )&=\{(u, R)\in \mathcal {U}\times [0,\infty )\quad |\quad (F_{4,R}), \lim _{s\rightarrow -\infty } u(s,t)=\star ,[u]=0\}, \end{aligned}$$

The bracket denotes a class in $\pi _{2}(M,L)/\ker \mu$, the vanishing conditions ensure the compatibility of the homotopy class of u with the capping of its ends $\underline{x}$ or $\underline{y}$ in the usual way. The over-lined $\bar{\sigma }$ denotes the capping $\sigma$ with reversed orientation.

For a generic choice of $(H,J,\star )$, these moduli spaces are smooth manifolds. The transversality for spaces of strips with moving boundary condition can be treated using the arguments in  [20] as the moving boundary condition problem can be turned into a fixed boundary condition problem by the following transformation. Let u satisfy ($F_{i}$) and let $v(s,t)=(\phi ^{t\chi _{i}(s)})^{-1}(u(s,t))$ then v(s, t) has fixed boundary on L and satisfies the equation:

$$\begin{aligned} \frac{\partial v}{\partial s}(s,t)- t\dot{\chi }_{i} Y^{t\chi _{i}(s)}(v(s,t)) + \tilde{J}(v(s,t))\Big (\frac{\partial v}{\partial t}(s,t)-\chi _{i}(s)Y^{t\chi _{i}(s)}(v(s,t)\Big )=0 \end{aligned}$$

(2)

where $-Y^{t}$ is the Hamiltonian vector field generating the Hamiltonian isotopy $\{(\phi ^{t})^{-1}\}$ and $\tilde{J}= D(\phi ^{t\chi _{i}(s)})^{-1}J D\phi ^{t\chi _{i}(s)}$.

However, the key argument in the proof of transversality, of being “somewhere injective” fails for the constant maps contained in the last moduli space $\mathcal {M}(\star ,\varnothing )$ (obtained for $R=0$).

Proposition 2.1

Let $\pi :\mathcal {M}(\star ,\varnothing )\rightarrow \mathbb {R}$ denote the projection. For $R = 0$, $\pi ^{-1}(R)$ consists in the single point $(u_{\star }, 0)$ where $u_{\star }$ is the constant map at $\star$. Moreover, this solution is regular, which means that (in the suitable functional spaces) the equation defining the moduli space $\mathcal {M}(\star ,\varnothing )$ is a submersion at this point.

Démonstration

For a map $(u,R) \in \mathcal {M}(\star ,\varnothing )$ the map $(\phi ^{\chi _{4,R}(s)t})^{-1}(u(s,t))= v(s,t)$ has fixed boundary conditions on L and satisfies Eq. 2. Notice that we can reformulate the problem in terms of maps from $u:(\bar{D}, \partial \bar{D}) \rightarrow (M, L)$ in the trivial homology class, where $\bar{D}$ denotes the unit disc and $\partial \bar{D} = S^{1}$. The condition $\lim \nolimits _{s\rightarrow -\infty }u(s,t)=\star$ is then replaced by $u(i) = \star$.

For $R = 0$ we have $(\phi ^{\chi _{4,R}(s)t})^{-1}\equiv Id$ and equation ($F_{4,R}$) is the perturbed non-linear Cauchy Riemann equation:

$$\begin{aligned} \frac{\partial u}{\partial s} + J_{t} (u)\frac{\partial u}{\partial t}=0. \end{aligned}$$

(3)

Points of $\mathcal {M}(\star , \varnothing )$ lying above $R = 0$ are hence discs (with boundary condition on L) in the trivial homology class satisfying (3) and are therefore constant. The additional condition $u(i) = \star$ implies $\pi ^{-1}(0) = \star$. The linearization (with respect to u) of the left hand term in (3) at the constant map $u_{\star }$ leads to a linear operator F with real boundary condition, defined for maps $\xi :(\bar{D}, \partial \bar{D})\rightarrow (T_{\star }M, T_{\star }L)$ of the form

$$\begin{aligned} F\xi =\partial _{s}\xi + J_{t} \partial _{t} \xi , \end{aligned}$$

(4)

where $J_{t} \in \text {Aut}(T_{\star }M)$ is a path of almost complex structures. To conclude that the kernel of F consists of constant maps we use the following argument from [21]: if $\xi \in \text {Ker}F$ then

$$\begin{aligned} \frac{1}{2}\int \limits _{\bar{D}} \Vert \partial _{s}\xi \Vert ^2_{J_{t}} + \Vert \partial _{s}\xi \Vert ^2_{J_{t}}dsdt=\int \limits _{\bar{D}} \omega _{\star }(\partial _{s}\xi ,J_{t}\partial _{s}\xi )dsdt =\int \limits _{\bar{D}} \xi ^{*}\omega _{\star }= \omega _{\star }([\xi ]) = 0 \end{aligned}$$

since $\pi _{2}(T_{\star }M,T_{\star }L)=0$ and hence $\xi$ is constant. The kernel of F is therefore n-dimensional and since n is also the index of F (because F is homotopic to the standard Cauchy Remann operator), this implies that F is surjective, which easily implies the required submersion property. $\square$

The dimension depends on the ends in the following way:

$$\begin{aligned} \dim \mathcal {M}(y,x)&=|y|-|x|\\ \dim \mathcal {M}(x,\varnothing )&=|x|\\ \dim \mathcal {M}(\star ,x)&=-|x|\\ \dim \mathcal {M}(\star ,\varnothing )&=1\\ \end{aligned}$$

These spaces are compact up to breaks at intermediate intersection points or bubbling of discs. However, the condition $\omega _{|\pi _{2}(M,L)}=0$ guarantees that there are no non-constant holomorphic discs in M with boundary on L. For monotone Lagrangian, bubbling may occur, either “on the side” of the strip, or the strip itself might break in the area where the hamiltonian term is turned off. The condition $N_{L}\ge 3$ is enough to prevent side bubbling on moduli spaces up to dimension 2, but the second phenomenon may still occur on the moduli space $\mathcal {M}(\star ,\varnothing )$, leading to configurations that consist of an holomorphic disc through $\star$ in some homotopy class $\alpha$, followed by a Floer strip where the hamiltonian term is turned off near both ends in the homotopy class $-\alpha$. The existence of such configurations is forbidden by the condition $N_{L}> \text {dim}(L)+1$.

Fig. 2

Two consecutive steps, one through an index 1 intersection, one through $\star$

All the moduli spaces of interest to us are hence compact up to breaks at intermediate intersection points, and there is a gluing construction ensuring that each broken configuration does indeed appear on the boundary of some moduli space.

In particular, given some $x\in \tilde{\mathcal {I}}_{0}(L,L_{1})$ and $y\in \tilde{\mathcal {I}}_{1}(L,L_{1})$, each broken trajectory $(\beta ,\alpha )\in \mathcal {M}(y,x)\times \mathcal {M}(x,\varnothing )$ belongs to the boundary of one component of $\mathcal {M}(y,\varnothing )$. Since this last space is 1-dimensional, it has to have another end $(\beta ',\alpha ')\in \mathcal {M}(y,x')\times \mathcal {M}(x',\varnothing )$ for some $x'\in \tilde{\mathcal {I}}_{0}(L,L_{1})$. We denote this relation between $(\beta ,\alpha )$ and $(\beta ',\alpha ')$ by

$$\begin{aligned} (\beta ,\alpha )\overset{\sharp }{\leftrightarrow }(\beta ',\alpha '). \end{aligned}$$

Definition 2.2

Given an index 0 intersection point $x \in \tilde{\mathcal {I}}_{0}(L,L_{1})$, a capping $\alpha \in \mathcal {M}(x,\varnothing )$ is called an “augmentation” of x, and the couple $(x, \alpha )$ an augmented orbit.

2.2 Steps and loops

Definition 2.3

A Floer step is an oriented connected component with non-empty boundary of a 1-dimensional moduli space $\mathcal {M}(y,\varnothing )$ for $y\in \tilde{\mathcal {I}}_{1}(L,L_{1})$ or $\mathcal {M}(\star ,\varnothing )$.

More explicitly, a Floer step in $\mathcal {M}(y,\varnothing )$ for $y\in \tilde{\mathcal {I}}_{1}(L,L_{1})$ can be identified to a quadruple $(\alpha _{0},\beta _{0},\beta _{1},\alpha _{1})$ where $\beta _{i}\in \mathcal {M}(y,x_{i})$ for some $x_{i}\in \tilde{\mathcal {I}}_{0}(L,L_{1})$, and $\alpha _{i}\in \mathcal {M}(x_{i},\varnothing )$ are such that

$$\begin{aligned} (\beta _{0},\alpha _{0})\overset{\sharp }{\leftrightarrow }(\beta _{1},\alpha _{1}). \end{aligned}$$

In $\mathcal {M}(\star ,\varnothing )$, exactly one special step

$$\begin{aligned} (\star ) \overset{\sharp }{\leftrightarrow }(\alpha _{\star },\beta _{\star }) \end{aligned}$$

has the constant J-holomorphic disc at $\star$ as one end (and a broken configuration $(\beta _{\star },\alpha _{\star })$ as the other end), while all the other steps can be described as a quadruple $(\alpha _{0},\beta _{0},\beta _{1},\alpha _{1})$ where $\beta _{i}\in \mathcal {M}(\star ,x_{i})$ for some $x_{i}\in \tilde{\mathcal {I}}_{0}(L,L_{1})$, and $\alpha _{i}\in \mathcal {M}(x_{i},\varnothing )$ are such that

$$\begin{aligned} (\beta _{0},\alpha _{0})\overset{\sharp }{\leftrightarrow }(\beta _{1},\alpha _{1}). \end{aligned}$$

With these notations, $\alpha _{0}$ is the start of the step, and $\alpha _{1}$ its end (Fig. 2).

Definition 2.4

A Floer based loop is a finite sequence of consecutive Floer steps starting and ending at $\star$. In other words, it is a sequence

$$\begin{aligned} \Big ((\star ,\beta _{\star },\alpha _{\star }),(\alpha _{\star }, \beta _{1},\beta '_{1},\alpha _{2}),(\alpha _{2},\beta _{2},\beta '_{2} ,\alpha _{3}),\dots , (\alpha _{N},\beta _{N},\beta '_{N},\alpha _{\star }),(\alpha _{\star }, \beta _{\star },\star )\Big ) \end{aligned}$$

such that $\forall i, (\beta _{i},\alpha _{i})\overset{\sharp }{\leftrightarrow }(\beta '_{i},\alpha _{i+1})$.

The set of all Floer loops is denoted by $\tilde{{\mathcal {L}_{\star }}}(L,L_{1})$.

Notice that $\tilde{{\mathcal {L}_{\star }}}(L,L_{1})$ depends on all the auxiliary data $(H, \star , J,\chi )$ but the dependency on the almost complex structure J and the cut off function $\chi$ is kept implicit to reduce the notation. $\tilde{{\mathcal {L}_{\star }}}(L,L_{1})$ carries an obvious concatenation rule that turns it into a semi-group. It also carries obvious cancellation rules. More explicitly, if $\sigma = (\alpha , \beta , \beta ' , \alpha ')$ is a Floer step, define its inverse $\sigma ^{-1}$ to be the same step with the opposite orientation: $\sigma ^{-1} = (\alpha ', \beta ', \beta , \alpha )$.

Denote by $\sim$ the associated cancellation rules in $\tilde{{\mathcal {L}_{\star }}}(L,L_{1})$. The concatenation then endows the quotient space

$$\begin{aligned} {\mathcal {L}_{\star }}(L,L_{1})= \tilde{{\mathcal {L}_{\star }}}(L,L_{1})\diagup \sim \end{aligned}$$

with a group structure.

Picking a parametrisation by [0, 1] of all the relevant components of the different moduli spaces, it also carries an evaluation map to the based loop space of L. More precisely, recall that each strip u in $\mathcal {M}(y,\varnothing )$ or $\mathcal {M}(\star ,\varnothing )$ converges to a point in L as s goes to $+\infty$ denoted by $u(+\infty )$ ; given a parametrisation $[0,1]\rightarrow \mathcal {M}(y,\varnothing )$ of a connected component $\mathcal {M}$ of such a moduli space, the map

$$\begin{aligned} \begin{array}{ccc} \mathcal {M}&{} \rightarrow &{} L\\ u &{}\mapsto &{}u(+\infty ) \end{array} \end{aligned}$$

defines a path in L. Concatenating the paths associated to all the steps , we get a loop in L based at $\star$:

$$\begin{aligned} {\mathcal {L}_{\star }}(L,L_{1})/_{\sim } \xrightarrow {\tilde{\mathrm {ev}}} \mathop {\Omega }(L,\star )\xrightarrow {\pi }\pi _{1}(L,\star ). \end{aligned}$$

The main statement of this section is then the following:

Theorem 2.5

If L is weakly exact or monotone with $N_{L}> \text {dim}(L) + 1$, then the evaluation map $\mathrm {ev}= \pi \circ \tilde{\mathrm {ev}}$:

$$\begin{aligned} {\mathcal {L}_{\star }}(L,L_{1}) \xrightarrow {\mathrm {ev}}\pi _{1}(L,\star ) \end{aligned}$$

is onto.

In other words, if $\sim$ is the homotopy equivalence relation in ${\mathcal {L}_{\star }}(L,L_{1})$, then:

Corollary 2.6

The map ${\mathcal {L}_{\star }}(L,L_{1})/_{\sim }\xrightarrow {\mathrm {ev}} \pi _{1}(L,\star )$ is an isomorphism.

The group ${\mathcal {L}_{\star }}(L,L_{1})/_{\sim }$ is called the Floer fundamental group of the pair $(L,L_{1})$ and it is denoted by

$$\begin{aligned} \pi _{1}(L, L_{1})={\mathcal {L}_{\star }}(L,L_{1})/_{\sim }. \end{aligned}$$

For a more economical presentation of the relations, but that uses an auxiliary Morse function, we refer to [1] where the proposed description should adapt straightforwardly.

The idea of the Proof of theorem 2.5 is to associate, to each Morse loop, an homotopic Floer loop (see, Proposition 2.7). Since it is well known that Morse loops generate the fundamental group, this is enough to obtain Theorem 2.5 (see, Proof in Sect. 2.3.1).

2.3 From Morse to Floer loops

Pick a Morse function f on L with a single minimum at $\star$ and a Riemannian metric g on L. The unstable and stable manifolds of a critical point a of f are denoted by $W^{u}(a)$ and $W^{s}(a)$, respectively. Assume f, g are both chosen generically so that all the Morse and PSS moduli spaces are defined transversely. Namely, we are interested in the following moduli spaces:

$$\begin{aligned} \mathcal {M}(b,a)&=(W^{u}(b)\cap W^{s}(a))/\mathbb {R}, \text { for } a,b\in \mathrm {Crit}(f),\\ \mathcal {M}(b,x)&=\{u\in \mathcal {M}(\varnothing ,x), u(-\infty )\in W^{u}(b)\}, \text { for } b\in \mathrm {Crit}(f), x\in \tilde{\mathcal {I}}(L,L_{1}),\\ \mathcal {M}(y,a)&=\{u\in \mathcal {M}(y,\varnothing ), u(+\infty )\in W^{s}(a)\}, \text { for } y\in \tilde{\mathcal {I}}(L,L_{1}), a\in \mathrm {Crit}(f). \end{aligned}$$

A Morse step is the path obtained by travelling once in one direction along the unstable manifold of an index 1 critical point (notice that since the Morse function is supposed to have only one index 0 critical point, Morse steps are also loops in this case) and Morse loops are finite sequences of Morse steps.

The space of Morse loops

$$\begin{aligned} {\mathcal {L}_{\star }}(f)=\langle \mathrm {Crit}_{1}(f)\rangle \end{aligned}$$

is then the free group generated by the index 1 critical points of f. For more details about this group see, [1]. Its elements encode sequences of oriented loops around the unstable manifolds of index one critical points, and hence have a realization as loops in L:

$$\begin{aligned} {\mathcal {L}_{\star }}(f)\xrightarrow {\tilde{\mathrm {ev}}}\mathop {\Omega }(L,\star )\xrightarrow {\pi } \pi _{1}(L,\star ), \end{aligned}$$

and we write $\mathrm {ev}= \pi \circ \tilde{\mathrm {ev}}$.

Let b be an index 1 critical point of f. Denote by $\gamma _{\pm }$ the two Morse flow lines rooted at b:

$$\begin{aligned} \mathcal {M}(b,\star )=\left\{ \gamma _{-},\gamma _{+}\right\} . \end{aligned}$$

Consider the space

$$\begin{aligned} B_{b}=\bigcup _{\begin{array}{c} y\in \tilde{\mathcal {I}}_{1}(L,L_{1})\cup \{\star \}\\ x\in \tilde{\mathcal {I}}_{0}(L,L_{1}) \end{array}} \mathcal {M}(b,y)\times \mathcal {M}(y,x)\times \mathcal {M}(x,\varnothing ) \bigcup \{(\gamma _{-},\star ),(\gamma _{+},\star )\}, \end{aligned}$$

of twice broken or degenerate hybrid Morse and Floer trajectories from b to $\varnothing$.

The “upper gluing” map $\sharp ^{\bullet }$ is defined as

$$\begin{aligned} \sharp ^{\bullet }: \begin{array}{ll} B_{b} &{} \rightarrow B_{b}\\ (\gamma ,\beta ,\alpha ) &{} \mapsto (\gamma ',\beta ',\alpha ) \hbox { such that }(\gamma ,\beta )\overset{\sharp }{\leftrightarrow }(\gamma ',\beta ')\\ (\gamma _{\pm },\star )&{} \mapsto (\gamma _{\mp },\star ), \end{array} \end{aligned}$$

here $(\gamma ,\beta )\overset{\sharp }{\leftrightarrow }(\gamma ',\beta ')$ means that $(\gamma ,\beta ), (\gamma ',\beta ')$ are the two ends of a connected component of a one-dimensional moduli space of the form $\mathcal {M}(b,x)$.

The “lower gluing” map $\sharp _{\bullet }$ is defined as

$$\begin{aligned} \sharp _{\bullet }: \begin{array}{ll} B_{b} &{} \rightarrow B_{b}\\ (\gamma ,\beta ,\alpha ) &{} \mapsto (\gamma ,\beta ',\alpha ') \hbox { such that }(\beta ,\alpha )\overset{\sharp }{\leftrightarrow }(\beta ',\alpha ')\\ (\gamma _{\pm },\star )&{} \mapsto (\gamma _{\pm },\beta _{\star },\alpha _{\star }) \end{array}. \end{aligned}$$

Both $\sharp ^{\bullet }$ and $\sharp _{\bullet }$ are one to one maps and moreover they are involutions. We refer to alternating iteration of $\sharp ^{\bullet }$ and $\sharp _{\bullet }$ as running a “crocodile walk” on $B_{b}$.

Since $B_{b}$ is finite and $\sharp ^{\bullet },\sharp _{\bullet }$ are involutions, all the orbits of the crocodile walk are periodic. Notice that the two degenerate configurations $(\gamma _{-},\star )$ and $(\gamma _{+},\star )$ are in the same orbit. Let

be the orbit containing the degenerate configurations.

The lower parts of these broken trajectories form a Floer loop

$$\begin{aligned} \psi (b)=\big ( (\star ,\beta _{0},\alpha _{0}),(\alpha _{0},\beta _{0},\beta _{1},\alpha _{1}), \dots , (\beta _{N},\alpha _{N},\star )\big ). \end{aligned}$$

Extending this map by concatenation, we get a group morphism

$$\begin{aligned} {\mathcal {L}_{\star }}(f)\xrightarrow {\psi }{\mathcal {L}_{\star }}(L,L_{1}). \end{aligned}$$

Consider now the following diagram

(5)

Proposition 2.7

The diagram  5 is commutative, i.e., for all $\gamma \in {\mathcal {L}_{\star }}(f)$, $\gamma$ and $\psi (\gamma )$ are homotopic.

Proof

Let b be an index 1 Morse critical point and consider the orbit of the crocodile walk introduced above to define $\psi (b)$.

Consider a trajectory (u, v) in a space of the form $\mathcal {M}(b,y)\times \mathcal {M}(y,\varnothing )$ for some $y\in \tilde{\mathcal {I}}_{1}(L,L_{1})\cup \{\star \}$ (or $\mathcal {M}(b,x)\times \mathcal {M}(x,\varnothing )$ for some $x\in \tilde{\mathcal {I}}_{0}(L,L_{1})$).

Evaluation of u and v along $\mathbb {R}\times \{0\}$ turns them into paths defined on $\mathbb {R}$. Parameterising their Morse part by the value of f and the Floer parts by the (suitable version of) the action turns them into Moore paths, and after concatenation, we get a Moore path in L rooted at b. This defines a map

$$\begin{aligned} \mathcal {M}(b,y)\times \mathcal {M}(y,\varnothing )\rightarrow \mathcal {P}(L,b), \end{aligned}$$

where $\mathcal {P}(L,b)$ is the space of Moore paths in L rooted at b. Notice this map is compatible with the compactification of the moduli spaces and extends continuously to their boundary.

The same holds for spaces of the form $\mathcal {M}(b,x)\times \mathcal {M}(x,\varnothing )$ for $x\in \tilde{\mathcal {I}}_{0}(L,L_{1})$ or the Morse space $\mathcal {M}(b,\varnothing )$, and hence, each upper or lower gluing in the crocodile walk defines a continuous one-parameter family of paths rooted at b, and whose end follows the evaluation of the step associated to that gluing. Moreover, along the orbit of the crocodile walk, each such family ends where the next one starts, so that they globally form a continuous $S^{1}$ family of rays, all rooted at b and whose other end describes the loop associated to that orbit.

This means that the orbit of the crocodile walk defines a trivial loop. Notice that one of the steps in this orbit is nothing but $\gamma _{+}\overset{\sharp }{\leftrightarrow }\gamma _{-}$, whose evaluation is the Morse loop associated to b, while the evaluation of the remaining part is $\mathrm {ev}(\psi (b))$. As consequence, $\mathrm {ev}(\psi (b))$ is homotopic to the Morse step associated to b.

Repeating this for each index 1 Morse critical point b, we obtain the result.

2.3.1 Proof of theorem 2.5

Theorem 2.5 is a straightforward corollary of proposition 2.7. Recall ${\mathcal {L}_{\star }}(f) \xrightarrow {\mathrm {ev}} \pi _{1}(L,\star )$ is onto. To see this, one can pick a representative $\gamma$ of an homotopy class $[\gamma ]\in \pi _{1}(L,\star )$. Using a genericity condition, one can suppose that $\gamma$ does not meet any stable manifold of codimension at most 2, and meets the stable manifolds of the index 1 critical points transversely. Pushing $\gamma$ down by the gradient flow of the Morse function then defines a homotopy from $\gamma$ to a Morse loop.

Remark 6

When $\gamma$ is a Floer loop, this process can be interpreted in terms of suitable moduli spaces and gluings, and fits exactly in the same formalism as the one used to move a Morse loop into a Floer loop.

Since ${\mathcal {L}_{\star }}(f)\xrightarrow {\mathrm {ev}}\pi _{1}(L,\star )$ is onto and the diagram (5) commutative, ${\mathcal {L}_{\star }}(L,L_{1})\xrightarrow {\mathrm {ev}} \pi _{1}(L,\star )$ has to be onto as well.

3 Fundamental group of Lagrangians with cylindrical ends

We now turn to Lagrangian submanifolds with cylindrical ends in $(\mathbb {C}\times M,\omega _{\mathrm {std}}\oplus \omega )$ as defined in [7].

Let W be a Lagrangian submanifold in $\mathbb {C}\times M$ with $N_{-}$ negative cylindrical ends $(L^{-}_{1},\dots ,L^{-}_{N_{-}})$, and $N_{+}$ positive ones $(L^{+}_{1},\dots ,L^+_{N_{+}})$: this means that there exist some real number A and collections of distinct constants $a^{\pm }_{1},\dots ,a^{\pm }_{N_{\pm }}$, such that

$$\begin{aligned} E_{\pm }(A):=W\cap \pi ^{-1}([\pm A,\pm \infty )\times \mathbb {R}) = \bigsqcup _{i=1}^{N_{\pm }}\Big ([\pm A,\pm \infty )\times \{a^{\pm }_{i}\}\Big )\times L^{\pm }_{i} \end{aligned}$$

and

$$\begin{aligned} W\cap \pi ^{-1}([-A,A]) \text { is compact}. \end{aligned}$$

The submanifold W is supposed to satisfy $\tilde{\omega }(\pi _{2}(\tilde{M},W))=0$ or to be monotone with minimal Maslov number $N_{W}> \text {dim}(W) + 1$.

3.1 Admissible Hamiltonian and almost complex structure

The previous construction requires some adaptation to make sense in this non compact setting. We start with the same setting used by [7] to define the Floer homology for Lagrangian submanifolds with cylindrical ends. Namely, let

$$\begin{aligned} B=[-A_{0},A_{0}]\times [-A_{0}, A_{0}]\subset \mathbb {C}\end{aligned}$$

Fig. 3

The base point should be chosen between bottlenecks...

Fig. 4

... or above a crossing in a cylindrical end

be a compact region in the plane outside which W is cylindrical, and $U^{\pm }_{i}(\epsilon )\subset V^{\pm }_{i}(\epsilon )$ be two nested neighborhoods of the ends of the form (for some $A > A_{0}$):

$$\begin{aligned} V^{\pm }_{i}(\epsilon )&=\left\{ (x,y)\in \mathbb {C}, \pm x> A_{0}- \epsilon ,|y-a^{\pm }_{i}|<\epsilon \right\} \\ U^{\pm }_{i}(\epsilon )&=\left\{ (x,y)\in \mathbb {C}, \pm x> A,|y-a^{\pm }_{i}|<\frac{\epsilon }{2} \right\} . \end{aligned}$$

Biran and Cornea define a Floer homology for almost complex structures and Hamiltonian functions satisfying the following conditions:

• (H1) the (possibly time dependent) almost complex structure $\tilde{J}$ on $\mathbb {C}\times M$ is $\tilde{\omega }$-compatible and satisfies $\tilde{J}=i\oplus J$ outside $B\times M$ for some (possibly time dependent) $\omega$-compatible almost complex structure J on M.

• (H2) The Hamiltonian function $\tilde{H}:[0,1]\times \mathbb {C}\times M\rightarrow \mathbb {R}$ is supported over $B\cup \bigcup _{i}V^{\pm }_{i}(\epsilon )$ and is linear above the ends:

  $$\begin{aligned} \forall (x,y,m)\in U^{\pm }_{i}(\epsilon )\times M,\quad \tilde{H}(x,y,m)=\alpha ^{\pm }_{i}x+\beta ^{\pm }_{i}, \end{aligned}$$

The signs of the reals $\alpha ^{\pm }_{i}$ affect the version of the resulting homology (relative to an end or not). In our case, we restrict to the case where:

$$\begin{aligned}&\forall i\in \{1,\dots ,N_{-}\},\quad \alpha ^{-}_{i}>0\nonumber \\&\forall i\in \{1,\dots ,N_{+}\},\quad \alpha ^{+}_{i}<0. \end{aligned}$$

(6)

We also pick a base point $\star \in W$, and on top of the previous conditions, we add one more technical restriction to easily ensure the compactness of the moduli spaces involving this base point. This condition can take two forms, depicted on Figs. 3 and 4.

Base point between bottlenecks (Fig. 3): the first option is to require that the Hamiltonian function has what Biran and Cornea call a “bottleneck” ( [8, Section 3.2]) on each end.

This means that above $V^{\pm }_{i}(\epsilon ){\setminus } U^{\pm }_{i}(\epsilon )$, $\tilde{H}$ splits as $\tilde{H}(x,y,m) = h^{\pm }_{i}(x)+ f^{\pm }_{i}(m)$ where the function $h^{-}_{i}:[-A, -A_{0}]\rightarrow \mathbb {R}$ (resp. $h^{+}_{i}:[A_{0}, A]\rightarrow \mathbb {R}$) has exactly one critical point, which is a non-degenerate local maximum $x^{-}_{i}$ (resp. local minimum $x^{+}_{i}$).

In this case, we require that the base point is chosen between the bottlenecks, i.e. that:

$$\begin{aligned} \max _{i}( x_{i}^{-}) \le \mathfrak {R}(\pi (\star ))\le \min _{j}(x_{j}^{+}). \end{aligned}$$

(H3)

This restriction provides a “last crossing” over each cylindrical end, that holomorphic curves cannot cross and hence keeps their projection in the $\mathbb {C}$ factor in a bounded region. In fact, any Hamiltonian function with this property would be as good as the above prescribed bottlenecks.

Base point above a self intersection of an end (Fig. 4): a variation of the above assumption that also easily ensures compactness, consists in requiring only one bottleneck, but then take the base point in the fiber above this special crossing. More explicitly, suppose there is some $L=L^{-}_{i}\in \{L^{-}_{1},\dots ,L^{-}_{N_{-}}\}$ (resp. $\{L^{+}_{1},\dots ,L^{+}_{N_{+}}\}$) such that above the corresponding end $V^{-}_{i}{\setminus } U^{-}_{i}$ (resp. $V^{+}_{i}{\setminus } U^{+}_{i}$), the Hamiltonian function $\tilde{H}$ splits as $\tilde{H}(x,y,m) = h(x)+ f(m)$ where h has exactly one critical point which is a local maximum (resp. minimum ) at some point $x_{0}$.

The point $p=x_{0}+ia^{-}_{i}$ (resp. $p=x_{0}+ia^{+}_{i}$) is then a transverse intersection point of the two curves given by the projection in $\mathbb {C}$ of the cylindrical end associated to L and its image under the hamiltonian isotopy. In this setting, we pick the base point $\star$ in the fiber p:

$$\begin{aligned} \star \in \pi ^{-1}(p) = \{p\}\times L. \end{aligned}$$

(H3')

3.2 Moduli spaces and Floer loops

Let $W_{1}=\phi _{\tilde{H}}^{1}(W)$ be the image of W under the Hamiltonian isotopy associated to $\tilde{H}$. The set $\tilde{\mathcal {I}}(W,W_{1})$ is defined in the same way as before, and we are interested in the moduli spaces $\mathcal {M}(y,x)$, $\mathcal {M}(x,\varnothing )$, $\mathcal {M}(\star ,x)$, $\mathcal {M}(\star ,\varnothing )$ introduced in Sect. 2.1.

For a generic choice of the auxiliary data $(\tilde{H},\tilde{J},\star )$ they are smooth manifolds of the expected dimension, but due to the non-compact setting, their compactness requires special attention.

3.2.1 Compactness of the moduli spaces

The compactness of the moduli spaces of the form $\mathcal {M}(y,x)$ is discussed in [7, Section 4.2]. The compactness of the moduli space $\mathcal {M}(\star ,x)$ is discussed in details in [7, Section 5.2], and [8, Section 3.2, 3.3]. More precisely, in this work Biran and Cornea use a Morse function g on W with $g|_{U_{\pm }(\epsilon )}=\tilde{H}$ over the ends, and discuss the compactness of hybrid Floer–Morse moduli spaces $\mathcal {M}(x,a)$ or $\mathcal {M}(a,x)$ for $x\in \tilde{\mathcal {I}}(W,W_{1})$ and $a\in \mathrm {Crit}(g)$ given as:

$$\begin{aligned} \mathcal {M}(x,a)&=\{u\in \mathcal {M}(x,\varnothing ), u(+\infty )\in W^{s}(a)\}\\ \mathcal {M}(a,x)&=\{u\in \mathcal {M}(\varnothing ,x), u(-\infty )\in W^{u}(a)\}. \end{aligned}$$

The moduli spaces $\mathcal {M}(\star ,x)$ fit into the description above (picking a Morse function g that has a minimum at $\star$).

We are also interested in moduli spaces of the form $\mathcal {M}(x,\varnothing )$ or $\mathcal {M}(\star ,\varnothing )$ where the Morse constraint is removed.

It turns out that if the ends are not moved in the right direction, i.e., if (6) is not fulfilled then this causes the moduli spaces $\mathcal {M}(x,\varnothing )$ and $\mathcal {M}(\star ,\varnothing )$ to be not compact up to breakings any more. If g is “decreasing” at infinity (i.e., g(x) is decreasing with |x| for |x| large enough), then its negative flow pushes any point down to a compact set and hence to a critical point. As a consequence the space $\mathcal {M}(x,\varnothing )$ can be seen as the union $\bigcup _{a\in \mathrm {Crit}(f)}\mathcal {M}(x,a)$ and hence it is compact up to breakings (bubbles are not allowed due to the usual asphericity or dimensional arguments).

Proposition 3.1

If we assume the conditions $(H_{1})$, $(H_{2})$ and one of the conditions (H3) or (H3’), the moduli spaces $\mathcal {M}(\star ,\varnothing )$ and $\mathcal {M}(x,\varnothing )$ for $x\in W\cap W_{1}$ are compact.

Proof

The only case that remains to be discussed is that of $\mathcal {M}(\star ,\varnothing )$.

Recall from [7, Section 4.2] that it is enough to show that the projection to the $\mathbb {C}$ factor of the curves in $\mathcal {M}(\star ,\varnothing )$ are bounded. Energy bounds and hence compactness follow easily.

Let us prove that in both settings, the projection to the $\mathbb {C}$ factor of the curves in $\mathcal {M}(\star ,\varnothing )$ do not enter any unbounded component of $\mathbb {C}{\setminus } (W\cup W_{1}\cup B)$.

Consider the setting given by assumption (H3), i.e., where the Hamiltonian function $\tilde{H}$ has “bottlenecks” above every end and the base point is chosen between them. If $\pi (u)$ enters a component that is not covered by $\pi (\cup _{t}\phi ^{t}_{\tilde{H}}(W))$ (see, Fig. 5), the maximum principle implies that it has to cover it all, and hence that u cannot have finite energy.

Fig. 5

A holomorphic disc cannot “go through” a crossing without entering forbidden sectors

On the other hand, an unbounded component that is covered by $\pi (\cup _{t}\phi ^{t}_{\tilde{H}}(W))$ is the strip delimited by the projection of a cylindrical end $(-\infty , x^{-}_{i}]\times \{a^{-}_{i}\}$ (or $[x^{+}_{i},+\infty )\times \{a^{+}_{i}\}$) and its image under $\phi ^{1}_{\tilde{H}}$. Since $\pi (\star )$ does not belong to such a component, if $\pi (u)$ does enter this component, then $\pi (u)$ has to go through the crossing point $z^{\pm }_{i}=(x^{\pm }_{i}+ia^{\pm }_{i})$, while avoiding two of the 4 sectors of $\mathbb {C}{\setminus }(\pi (W\cup W_{1}))$ near $z^{\pm }_{i}$. This implies that $\pi (u)$ has to be constant, and hence that u is contained in the fiber $\pi ^{-1}(z^{\pm }_{i})$. This contradicts the assumption that $\pi (u)$ entered the unbounded component.

The same argument applies to the second setting, given by assumption (H3’), where the base point is chosen in the fiber of a self crossing of a cylindrical end. In this situation, for all $(u,R)\in \mathcal {M}(\star ,\varnothing )$, $\pi (u)$ has to go through $p=\pi (\star )$ while avoiding two of the 4 sectors of $\mathbb {C}{\setminus }(W\cup W_{1})$ near $\pi (\star )$. Recall from  [7] that this implies it is constant, and hence that all the curves in $\mathcal {M}(\star ,\varnothing )$ are contained in the fiber $\pi ^{-1}(z^{\pm }_{i_{0}})$. In particular, their projection is bounded.

Remark 7

(Genericity of assumption(H3’)) The condition $\pi (\star )=p$ is obviously not a generic one. However, it still offers enough freedom to ensure transversality, i.e., that all the relevant moduli spaces are cut out transversely. In fact, above we proved that all the trajectories we are interested are contained in the single fiber $M=\pi ^{-1}(p)$ using similar ideas as in [7, Section 4.2]. A generic choice of $\star$ (and all the other auxiliary data) on L ensures transversality for the relevant moduli spaces of curves in M. To ensure transversality for the same curves seen in $\mathbb {C}\times M$, we need to check that the linearized operators $\bar{\partial }_{u}$ (with its associated boundary conditions) at each curve u seen in $\mathbb {C}\times M$ are onto.

But for the chosen structure, $\bar{\partial }_{u}$ splits as a direct sum $\bar{\partial }\oplus \bar{\partial }_{u_{M}}$ where the first $\bar{\partial }$ is associated to the constant curve $\{p\}$ and boundary conditions associated to $T_{p}\pi (W)$ and $T_{p}\pi (W_{1})$. Then the linearized operator associated to $\bar{\partial }$ is also surjective by the same argument that in Proposition 2.1 (its kernel consists of the real constant maps and then its Index is equal to the dimension of its Kernel). By assumption the linearization of $\bar{\partial }_{u_{M}}$ is surjective, therefore the operator corresponding to the linearization of $\bar{\partial }\oplus \bar{\partial }_{u_{M}}$ is surjective.

As consequence, the notions of loops given in 2.3 make sense in this non-compact setting.

3.3 Homotopies

We fix a Morse function f (and a Riemannian metric $\rho$) on W with a single minimum at $\star$, and such that $f=\tilde{H}$ up to a constant over the ends.

The restriction (6) ensures that the negative flow of the Morse function pushes any loop in W down to a compact set and hence to the 1-skeleton given by f: this procedure defines a morphism (see, definition in [1, Section 3.1])

$$\begin{aligned} {\mathcal {L}_{\star }}(L,L_{1}) \xrightarrow {\phi } {\mathcal {L}_{\star }}(f), \end{aligned}$$

that still makes sense in this non-compact setting.

Similarly, once all the relevant moduli spaces are proven to be compact up to breaking, the crocodile walk can be run just like in the compact case, and the morphism

$$\begin{aligned} {\mathcal {L}_{\star }}(f) \xrightarrow {\psi } {\mathcal {L}_{\star }}(L,L_{1}), \end{aligned}$$

defined in 2.3 is still well defined in this non-compact setting. The diagram (5) still holds with the same commutative property.

Moreover, the required behavior of f over the ends ensures that the Morse loops do indeed generate the fundamental group despite of the non-compactness of W. As a consequence, the Floer loops do also generate the fundamental group.

Theorem 3.2

${\mathcal {L}_{\star }}(W,W_{1})\xrightarrow {\mathrm {ev}}\pi _{1}(W)$ is onto.

4 Cobordisms with two ends

We now restrict attention to the case when W has exactly one negative end $L_{-}$ and one positive end $L_{+}$ (and still satisfies the restriction $\tilde{\omega }(\pi _{2}(\tilde{M},W))=0$ or it is monotone with $N_{W}> \text {dim}(W) + 1$).

The goal of this section is to prove Theorem 1.2. To this end, we use suitable auxiliary data $(\tilde{H},\tilde{J},\star )$ to compare $\pi _{1}(W)$ and $\pi _{1}(L_{-})$.

Namely, a first Hamiltonian $\tilde{H}$ is chosen, satisfying condition (H2) in Sect. 3.1 and such that the associated isotopy $(\Phi ^{t})$ moves $W\cap \pi ^{-1}([-A_{0},A_{0}]\times \mathbb {R})$ far enough downward (in the y-direction) in the $\mathbb {C}$ factor to separate it from itself, while it translates the cylindrical ends vertically, upward for the negative end and downward for the positive one: for $a_{-}, a_{+}, A, K \in \mathbb {R}$ with $a_{-} > 0$ and $a_{+} < 0$ we have $\pi (\Phi ^{1}((-\infty , A]\times \{0\}\times L_{-})) = (-\infty , A\pm K ]\times \{a_{-}\}$ and $\pi (\Phi ^{1}([A,\infty )\times \{0\}\times L_{+})) = [A \pm K,\infty )\times \{a_{+}\}$.

Notice that in this situation, all the intersections of W and $W_{1}$ lie above a single point $p\in \mathbb {C}$, that belongs to the area where the projection $\pi :\mathbb {C}\times M\rightarrow \mathbb {C}$ is holomorphic and where $\pi (W)$ and $\pi (W_{1})$ are two transverse curves.

We pick the base point $\star$ in the fiber $\{p\}\times L_{-}$ above p [condition (H3’)].

The maximum principle used in [7] and in the proof of proposition 3.1 above then implies that for all $x,y\in \tilde{\mathcal {I}}(W,W_{1})$:

$$\begin{aligned}&\forall u\in \mathcal {M}(y,x),\quad \pi (u(\mathbb {R}\times [0,1]))=\{p\},\\&\forall u\in \mathcal {M}(x,\varnothing ),\quad \pi (u(\mathbb {R}\times [0,1]))=\{p\},\\&\forall u\in \mathcal {M}(\star ,\varnothing ),\quad \pi (u(\mathbb {R}\times [0,1]))=\{p\}. \end{aligned}$$

4.1 Proof of Theorem 1.2

Let $\big (\phi ^{t} =(\phi _{\mathbb {C}}^{t}, \phi ^{t}_{M})\big )$ be the Hamiltonian isotopy associated with $\tilde{H}$ and $L_{1}=\phi ^{1}_{M}(L_{-})=W_{1}\cap \pi ^{-1}(p)$. From the previous section, we derive that Floer steps in $(M,L_{-},L_{1})$ are the same as the Floer steps in $(\mathbb {C}\times M, W,W_{1})$.

Moreover, loops that are homotopically trivial in $L_{-}$ are obviously also trivial in W,

so that we get the following diagram:

This ends the proof that $i_{\sharp }$ is onto for $(W; L_{-}, L_{+})$ weakly exact or monotone with minimal Maslov number $N_{W}> dim(W) +1$.

Similarly, to prove that $\pi _{1}(L_{+},\star ) \xrightarrow {i_{\sharp }} \pi _{1}(W,\star )$ is onto one considers a perturbation $\tilde{H}$ satisfying conditions (H2) and $(H3')$ in Sect. 3.1 and such that the associated isotopy $(\phi ^{t})$ moves $W\cap \pi ^{-1}([-A_{0},A_{0}]\times \mathbb {R})$ far enough downward (in the y-direction) in the $\mathbb {C}$ factor to separate it from itself, while it translates the cylindrical ends vertically, upward for the positive end and downward for the negative one: for $a_{-}, a_{+}, A, K \in \mathbb {R}$ with $a_{-} < 0$ and $a_{+} > 0$ we have $\pi (\Phi ^{1}([A, \infty )\times \{0\}\times L_{+})) = [A\pm K, \infty ) \times \{a_{+}\}$ and $\pi (\Phi ^{1}(-\infty , A]\times \{0\}\times L_{-})) = (-\infty , A \pm K]\times \{a_{-}\}$. The rest of the proof is identical.

4.2 Proof of Corollary 1.3

Suppose moreover that $\pi _{2}(M,L_{-})=0$. From the homotopy long exact sequence of the pair $(M,L_{-})$ we derive that the inclusion of $L_{-}$ in M induces an injective group morphism $\pi _{1}(L_{-})\hookrightarrow \pi _{1}(M)$. This also holds for the inclusion of $L_{-}$ in $\mathbb {C}\times M$, and since it factors through the inclusion of $L_{-}$ in W, we conclude that the induced map $\pi _{1}(L_{-})\xrightarrow {i_{\sharp }}\pi _{1}(W)$ has to be injective.

This ends the proof of corollary 1.3.

4.2.1 Proof of Theorem 1.6

To prove this theorem we use a version of Floer homology called quantum homology. Quantum homology was initially defined in  [15, Section 5.1] for closed Lagrangians, monotone with minimal Maslov number at least two. In [7, Section 3.3] the definition was adapted to Lagrangian cobordism $(W;L,L')$. Denote by $QH(W,L;\mathbb {Z}[\pi _{1}(W)])$ the homology of the complex $\mathcal {C}(f, \rho , J;\mathbb {Z}[\pi _{1}(W)])= (\mathcal {C}_{*}(f, \rho , J), d_{*})$. This complex is a $\mathbb {Z}$-graded free $\mathbb {Z}[\pi _{1}(W)]$-chain complex³ defined for a triple $\mathcal {D}=(f, \rho , J)$ composed of a Morse function $f : \hat{W} \rightarrow \mathbb {R}$, (recall from definition 1.1 that $\hat{W}$ denotes the compact part of W) a Riemannian metric $\rho$ on W and an almost complex structure J. The data is assumed to be generic. We assume that the gradient vector field $-\nabla _{\rho }f$ is transverse to $\partial \hat{W}$, and it points outside along L and it points inside along $L'$. The function f is extended linearly to W.

The ring $\Lambda = \mathbb {Z}[t^{-1}, t]$ is graded by setting $\text {deg}(t) = -N_{W}$, then $\Lambda = \bigoplus \nolimits _{i\in \mathbb {Z}}\Lambda _{i}$, where $\Lambda _{i}$ is the subring of homogeneous elements of degree i. Denote by $\text {Crit}_{i}(f)$ the set of index i critical points of f. Then

$$\begin{aligned} \mathcal {C}_{*}(f, \rho , J):= \bigoplus \limits _{i \in \mathbb {Z}} \mathbb {Z}[\pi _{1}(W)]\otimes \mathbb {Z}\langle \text {Crit}_{* + iN_{W}}(f)\rangle \otimes \Lambda _{-iN_{W}}, \end{aligned}$$

and the differential $d:\mathcal {C}_{*}(f, \rho , J)\rightarrow \mathcal {C}_{*-1}(f, \rho , J)$ counts configurations in the moduli space $\mathcal {P}(x, y, A; \mathcal {D})$, of “(non parametrized) pearl trajectories from x to y in the class A” where $x, y \in \text {Crit}(f)$ and $A\in \pi _{2}(\tilde{M}, W)$.

Such a pearl trajectory can be written as a $(2k+1)$-tuple $\bar{u} = (\gamma _{0},u_{1},\gamma _{1},u_{2},\dots ,u_{k},\gamma _{k})$ where

• each $\gamma _{i}$ is a piece of Morse flow line,

• if $k>0$, each $u_{i}$ is a (non parametrized) non trivial holomorphic disc with boundary in L, with 2 marked points $(p^{-},p^{+})$ on the boundary,

such that

• $\gamma _{0}$ joins x to $u_{1}(p^{-})$,

• $\gamma _{i}$ is non constant and joins $u_{i}(p^{+})$ to $u_{i+1}(p^{-})$ for $1\le i \le k-1$,

• and $\gamma _{k}$ joins $u_{k}(p^{+})$ to y.

Here, “non parametrized” means that we quotient the space of pearl trajectories by the action of the automorphisms of each disc preserving the marked points.

To associate an element in $\pi _{2}(M,W)$ to such a pearl trajectory, we pick for each $z \in \text {Crit}(f)$ a path $\gamma _{z}$ from z to a fixed base point $\star \in W$, and denote by $\bar{\gamma _{z}}$ the path $\gamma _{z}$ with reversed orientation.

Then, given a pearl trajectory $\bar{u}$ from x to y as above, the concatenation of all the discs $u_{i}$, the Morse flow lines $\gamma _{i}$, and $\gamma _{y}$ defines a class $[\bar{u}]\in \pi _{2}(M,W)$. The space $\mathcal {P}(x, y, A; \mathcal {D})$ is obtained by requiring that

$$\begin{aligned}{}[\bar{u}]=A. \end{aligned}$$

The dimension of $\mathcal {P}(x, y, A; \mathcal {D})$ is given by $|x|-|y| + \mu (A)- 1$. For a proof of the transversality of these spaces see, [14, Proposition 3.1.3].

Each pearl trajectory $\bar{u}$ from x to y defines a path $\partial _{-}(\bar{u})$ obtained by following the flow lines and turning from $p_{-}$ to $p_{+}$ counterclockwise on the boundary of each disc. We let $g_{\overline{u}}$ denote the homotopy class of the loop obtained by concatenation of $\bar{\gamma _{x}}\# \partial _{-}\overline{u} \# \gamma _{y}$.

The differential is then given by the following expression from  [22, Apendix A.2.1]:

$$\begin{aligned} d(x) := \sum \limits _{\begin{array}{c} y\in \text {Crit}(f)\\ |x|-|y| = 1 \end{array}} \sum \limits _{u\in \mathcal {P}(x, y, 0; \mathcal {D})} g_{u}y +\sum \limits _{\begin{array}{c} A \in \pi _{2}(\tilde{M},W), y\in \text {Crit}(f)\\ |x|-|y| + \mu (A) = 1 \end{array}} \sum \limits _{u\in \mathcal {P}(x, y, A; \mathcal {D})} (-1)^{|y|}g_{u}y t^{\mu (A)/N_{W}}. \end{aligned}$$

The differential satisfies $d^{2} = 0$: this is a straightforward adaptation of the proof given in [22, Apendix A.2.1].

In the monotone case, our assumption that $N_{W}>\dim W+1$ implies that the condition $|x|-|y|+\mu (A)=1$ can only be fulfilled if $\mu (A)=0$ (because $|x|-|y|\ge -\dim W$). This condition in turn implies that the discs in such a pearl trajectory have vanishing symplectic area, and hence are trivial. As a consequence, for $0 \le * \le \text {dim}(W)$ we have that $\mathcal {C}_{*}(f,\rho , J)= \mathbb {Z}[\pi _{1}(W)]\otimes \mathbb {Z}\langle \text {Crit}_{*}(f)\rangle \otimes \Lambda _{0}$.

This means that $d_{*}= d_{Morse}$ is the Morse differential for $0 \le * \le \text {dim} W$. So the complex $\mathcal {C}(f, \rho , J;\mathbb {Z}[\pi _{1}(W)])$ contains a copy of the $\mathbb {Z}[\pi _{1}(W)]$-Morse complex of the pair (W, L):

$$\begin{aligned} \cdots \rightarrow C_{\text {dim}(W)+1}\left( f, \rho , J\right)&\xrightarrow {d_{\text {dim}(W)+1}} C_{\text {dim}(W)}\left( f, \rho , J\right) \rightarrow \cdots \rightarrow C_{0}(f, \rho , J)\xrightarrow {d_{0}}\ldots \end{aligned}$$

(7)

Let us denote

$$\begin{aligned} CM\left( W,L;\mathbb {Z}\left[ \pi _{1}\left( W\right) \right] \right) = \bigoplus \limits _{i=0}^{\text {dim(W)} }C_{i}(f,\rho , J). \end{aligned}$$

Recall that the $\mathbb {Z}[\pi _{1}(W)]$-Morse complex of the pair (W, L) is the complex associated to a Morse function on W with differential defined in the same way as d above but taking into account only the spaces $\mathcal {P}(x, y,A; \mathcal {D})$ with $A = 0$. By lifting the pair $(f, \rho )$ to the universal covering of W, denoted $\tilde{W}$, we can define a cellular decomposition of the pair $(\tilde{W},p^{-1}(L))$, so $H(CM(W,L;\mathbb {Z}[\pi _{1}(W)]))\cong H(\tilde{W}, p^{-1}(L); \mathbb {Z})$. For $0 \le * \le \text {dim}(W)$ the subcomplex $(\mathcal {C}_{*}(f, \rho , J), d_{*})$ is precisely the $\mathbb {Z}[\pi _{1}(W)]$-Morse complex of the pair (W, L). For $N_{W}> \text {dim}(W) +1$ differentials $d_{\text {dim}(W)+1}$ and $d_{0}$ are both 0 (since the moduli spaces are empty by dimension reason). This implies that $CM(W,L;\mathbb {Z}[\pi _{1}(W)]) \hookrightarrow \mathcal {C}(f, \rho , J;\mathbb {Z}[\pi _{1}(W)])$ induces an injective map $H(CM(W,L;\mathbb {Z}[\pi _{1}(W)])) \rightarrow QH(W,L;\mathbb {Z}[\pi _{1}(W)])$. On the other hand, we know that $QH(W,L;\mathbb {Z}[\pi _{1}(W)]) = 0$ since the cobordism is displaceable [7, Theorem 2.2] and there exists a PSS-type isomorphism between the Floer homology with local coefficients and the quantum homology with local coefficients (see, [7, Proposition 5.2.1] for $\mathbb {Z}_{2}$-coefficients and [23, Theorem 1.3] for arbitrary coefficients). From this we obtain $H_{*}(CM(W,L;\mathbb {Z}[\pi _{1}(W)])) = H_{*}(\tilde{W}, p^{-1}(L); \mathbb {Z}) = 0$. By hypothesis $L\hookrightarrow W$ induces an injective map on the fundamental group, since we previously have proved that $\pi _{1}(L)\rightarrow \pi _{1}(W)$ is surjective, then we have the $\pi _{1}(L)\rightarrow \pi _{1}(W)$ is an isomorphism, so $p^{-1}(L) = \tilde{L}$. From the long exact sequence in homology of the pair $(\tilde{W},\tilde{L} )$ and one of the Whitehead Theorems we obtain that the inclusion $L\hookrightarrow W$ is a homotopy equivalence. We proceed in an analogous way to show that $L'\hookrightarrow W$ is a homotopy equivalence.

Remark 8

Alternatively, notice that from the proof above we have $H_{*}(\tilde{W}, p^{-1}(L); \mathbb {Z}) = 0$, so in particular $H_{0}(p^{-1}(L); \mathbb {Z}) \cong H_{0}(\tilde{W}; \mathbb {Z}) \cong \mathbb {Z}$. As pointed out to us by Baptiste Chantraine (see, Remark 2) this implies that $\pi _{1}(L)\rightarrow \pi _{1}(W)$ is surjective.