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.
Categories: math.SG