Continuation maps for the Morse fundamental group

We study properties of the continuation map for the Morse fundamental group [Formula: see text] associated to a Morse–Smale pair [Formula: see text] on a manifold [Formula: see text]. We get a morphism between [Formula: see text] and [Formula: see text] 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 [Formula: see text] 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 [Formula: see text].