Congruent zeta functions. No.13

Yoshifumi Tsuchimoto

Let us site wikipedia:

(https://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem with some modifications by Tsuchimoto-consult the original page for details)

THEOREM 13.1 (Lefschetz)   Let

$$f: X\rightarrow X$$

be a continuous map from a compact triangulable space $X$ to itself. Define the Lefschetz number $\Lambda_f$ of $f$ by

$$\Lambda_ f :=\sum_{k\geq 0}(-1)^k \operatorname{tr}(f_*\vert H_k(X,\mathbb{Q})),$$

the alternating (finite) sum of the matrix traces of the linear maps induced by $f$ on $H_{k}(X,\mathbb{Q} )$, the singular homology groups of $X$ with rational coefficients. A simple version of the Lefschetz fixed-point theorem states: if $\Lambda_f \neq 0$, then $f$ has at least one fixed point, i.e., there exists at least one $x$ in $X$ such that $f(x)=x$. In fact, since the Lefschetz number has been defined at the homology level, the conclusion can be extended to say that any map homotopic to $f$ has a fixed point as well.

A stronger form of the theorem, also known as the Lefschetz-Hopf theorem, states that, if $f$ has only finitely many fixed points, then

$$\sum_{x \in \mathrm{Fix}(f)} i(f,x) = \Lambda_f,$$

where $\mathrm {Fix} (f)$ is the set of fixed points of $f$, and $i(f,x)$ denotes the index of the fixed point $x$. From this theorem one deduces the Poincaré-Hopf theorem for vector fields