Every map from compact manifold are homotopic to map with finitely many fixed points

Question

Helo, i found this lemma without proof.

Lemma: Every map from compact manifold are homotopic to map with finitely many fixed points

Only tip that has been given - "by transversality argument". How can i proof this. I found partial proof when M is smooth manifold, but generaly i have no idea how to proof it for all compact manifolds.

Have a nice day, Adrian

Answer 1

Hint: The graph $G$ of $f$ is a subset of $M \times M$; consider how it's related to the diagonal $\Delta \subset M \times M$, which is the graph of the identity function $i$: a point of $G \cap \Delta$ corresponds to a fixed point of $f$. So if $f$ is transverse to $i$, then what can you say about the intersection?

Answer 2

I found this lemma in "Homotopy Methods in Topological Fixed and Periodic Point Theory - Jerzy Jezierski & Waclaw Marzantowicz" Lemma (4.2.3).

We can not say that the function is transverse because M is not differentiable, right?