Equivariant Morse Homology for Reflection Actions via Broken Trajectories
Erkao Bao, Tyler Lawson, Lina Liu · International Mathematics Research Notices · 2026
Abstract We consider a finite group $G$ acting on a manifold $M$. According to [3, 10], a generic equivariant function on $M$ is Morse. For any equivariant Morse function, 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 $\phi : M \to M$ is a reflection if $\phi ^{2} = \operatorname{id}$ and the fixed point set of $\phi $ 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 $G = \{\operatorname{id}, \phi \}$. 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 [7]. 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.