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.