TITLE: Morse functions invariant under diffeomorphisms
QUESTION [7 upvotes]: Let $f:M \to \mathbb{R}$ be a Morse function of a compact manifold $M$. Assume $\sigma:M \to M$ is a diffeomorphism such that $f$ is invariant under $\sigma$, i.e. $f(\sigma x)=f(x)$ for all $x \in M$.
I am trying to understand how $\sigma$ induces homomorphisms of the Morse homology groups $H_k(M)$ and how to describe them explicitly. If we denote the boundary maps by $\partial_k$, the elements of $H_k(M)$ are linear combinations of cosets of the form $x+\text{im }\partial_{k+1}$ where $x$  is a critical  point of index $k$ with $x \in \ker \partial_k$.
Question: Does $\sigma$ act on $H_k(M)$ by application to representatives of the cosets? That is: Is the map
$$x+\text{im }\partial_{k+1} \mapsto \sigma x+\text{im }\partial_{k+1}$$
well defined and does it extend to an automorphism of $H_k(M)$?
I think I am able to see that $\sigma$ permutes the critical points of a fixed index. This would show that $\sigma$ is an automorphism of the Morse chain groups $C_k(M)$. However, I don't know how to proceed. The standard approach would be to show that $\sigma$ commutes with the boundary maps $\partial_{k+1}$ but I have no ideas how to do this. I tried to analyze the action of $\sigma$ on the trajectories between critical points but up to now, this didn't result in anything useful for me. Not sure if this is a good strategy here...

REPLY [3 votes]: You have to be careful, Morse chain complexes are undefined unless you pick a Riemannian metric on the manifold, however, it will be true that $\sigma$ induces a degree preserving isomorphism on the Morse homology of $M$ (and it does not depend on the choice of this Riemannian metric).
Indeed, recall the following result which is standard in Morse theory:

Let $M$ and $N$ be closed manifolds, let $(f,g)$ be a Morse-Smale pair on $N$ and let $\sigma\colon M\to N$ be a diffeomorphism, then $(f\circ\sigma,\sigma_*g)$ is a Morse-Smale pair on $M$ and $\sigma$ induces a degree preserving isomorphism on the Morse chains complexes of $(f,g)$ and $(f\circ\sigma,\sigma_*g)$ and it descends to a degree preserving isomorphism between the Morse homologies of $M$ and $N$.

Sketch of a proof. Recall that Morse homology is defined in a way that

Chains. The chains are spanned by the critical points of the Morse function.
Grading. The grading is given by the index of the critical points of this Morse function.
Differential. The differential is given by a count of gradient trajectories of the Morse function.

Notice that the chain rule implies that $\sigma$ is a bijection between the critical points of index $k$ of $f$ and the critical points of index $k$ of $f\circ\sigma$, it is thus a bijection between $MC_k(f,g)$ and $MC_k(f\circ\sigma,\sigma_*g)$.
Now, what about the differential? The chain rule implies that $\nabla_{\sigma_*g}(f\circ\sigma)=\sigma^*\nabla_gf$ and yet another use of the chain rule shows that the flows of $\nabla_{\sigma_*g}(f\circ\sigma)$ and $\nabla_gf$ are conjugated by $\sigma$, meaning that if $\gamma$ is a $g$-gradient trajectory of $f$ joining $\sigma(q)$ to $\sigma(p)$, then $\sigma^{-1}\circ\gamma$ is the unique $\sigma_*g$-gradient trajectory of $f\circ\sigma$ joining $q$ to $p$. In short, $\sigma$ commutes with the Morse differential. $\Box$
In your case, $M=N$ and $f\circ\sigma=f$, therefore the previous result shows that $\sigma$ induces an isomorphism between $MC_k(M,f,g)$ and $MC_k(M,f,\sigma_*\varphi)$ and if $\sigma$ is furthermore an isometry of $g$, then $\sigma$ induces an automorphism of $MC_k(M,f,g)$.
The map you defined in your question is indeed a degree preserving automorphism in homology.