Let $X$ be a smooth projective variety. If I've understood correctly, the Weil conjectures imply that it is possible to compute the Betti numbers of $X(\mathbb{C})$ by computing the local zeta function associated to $X(\overline{\mathbb{F}_p})$ for $p$ a prime of good reduction. This is because one can identify the factors coming from different $\ell$-adic cohomology groups by the absolute value of their roots (the "Riemann hypothesis," by analogy with the case of curves).

Unfortunately, this doesn't seem to be true for the analogous situation with compact triangulable spaces $Y$ and the Lefschetz zeta function $\zeta_f$, at least not for an arbitrary choice of function $f : Y \to Y$. For example, if $f$ is homotopic to the identity, then $\zeta_f$ can only see the Euler characteristic of $Y$. Even if $f$ acts "generically" (i.e. none of the factors of $\zeta_f$ cancel), there doesn't seem to be a way to distinguish which factors are associated to which degree. (Of course, I would love if I were wrong about this.)

**Question 1:** When is there an analogue of the geometric Frobenius for compact triangulable spaces $Y$? By this I mean a more-or-less canonical function $f : Y \to Y$ such that some analogue of the Riemann hypothesis holds for $\zeta_f$.

**Question 2:** Regardless of the answer to Question 1, is it always possible to choose $f$ such that $\zeta_f$ can tell you which of its factors are associated to which homology groups?

(Side question: is it true that given any $f : Y \to Y$ there is always some $f'$ homotopic to $f$ which has finitely many fixed points?)