# Morse functions invariant under diffeomorphisms

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...

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.