Continuation maps for the Morse fundamental group

Abstract

We study properties of the continuation map for the Morse fundamental group $π_1^\text{Morse}(f,\ast)$ associated to a Morse-Smale pair $(f,g)$ on a manifold $M$. We get a morphism between $π_1^\text{Morse}(f_1,\ast_1)$ and $π_1^\text{Morse}(f_2,\ast_2)$ and show that it is functorial. We also define the morphism in the case of Morse data over different manifolds, thanks to the use of grafted trajectories. Finally, given an interpolation function on $M\times\mathbb{R}$ between two Morse functions (used for example to define the continuation map), we study the Morse fundamental group associated to that function and show that it is isomorphic to a relative fundamental group on $M\times\mathbb{R}$.