Equivariant Morse Homology for Reflection Actions via Broken Trajectories

Abstract

We consider a finite group $G$ acting on a manifold $M$. For any equivariant Morse function, which is a generic condition, there does not always exist an equivariant metric $g$ on $M$ such that the pair $(f,g)$ is Morse-Smale. Here, the pair $(f,g)$ is called Morse-Smale if the descending and ascending manifolds intersect transversely. The best possible metrics $g$ are those that make the pair $(f,g)$ stably Morse-Smale. A diffeomorphism $ϕ: M \to M$ is a reflection, if $ϕ^2 = \operatorname{id}$ and the fixed point set of $ϕ$ forms a codimension-one submanifold (with $M \setminus M^{\operatorname{fix}}$ not necessarily disconnected). In this note, we focus on the special case where the group $G = \{\operatorname{id}, ϕ\}$. We show that the condition of being stably Morse-Smale is generic for metrics $g$. Given a stably Morse-Smale pair, we introduce a canonical equivariant Thom-Smale-Witten complex by counting certain broken trajectories. This has applications to the case when we have a manifold with boundary and when the Morse function has critical points on the boundary. We provide an alternative definition of the Thom-Smale-Witten complexes, which are quasi-isomorphic to those defined by Kronheimer and Mrowka. We also explore the case when $G$ is generated by multiple reflections. As an example, we compute the Thom-Smale-Witten complex of an upright higher-genus surface by counting broken trajectories.