# The fundamental group of a rigid Lagrangian cobordism

## TL;DR

This paper extends Floer theory to monotone Lagrangians to analyze the fundamental group of Lagrangian cobordisms, revealing conditions under which the cobordism induces surjective and injective fundamental group maps.

## Contribution

It introduces a Floer fundamental group construction for monotone Lagrangians and applies it to study the fundamental group of Lagrangian cobordisms, establishing new surjectivity and injectivity results.

## Key findings

- Inclusions induce surjective maps on fundamental groups.
- Injectivity of these maps implies the cobordism is an h-cobordism.
- Extension of Floer fundamental group to the monotone setting.

## Abstract

In this article we extend the construction of the Floer fundamental group to the monotone Lagrangian setting 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.