Lefschetz fixed-point theorem

Remarks

The idea of the proof: suppose $f$ has no fixed point. By compactness $d(x, f(x))$ is bounded below, so after fine subdivision $f$ has a simplicial approximation that moves every simplex off itself. The chain map of this approximation then has zero trace in each degree. The Hopf trace formula equates the alternating sum of traces on chains with that on homology, so $L(f) = 0$. Example: if $X$ is contractible and compact as above, then $H_0(X; \Q) = \Q$ and the other groups vanish, so $L(f) = 1$ for every $f$ and every continuous self-map has a fixed point. For $f$ homotopic to the identity, $L(f) = \chi(X)$, the Euler characteristic; so if $\chi(X) \neq 0$, every map homotopic to the identity has a fixed point. Compactness is necessary: $\R$ is contractible, so $L(f) = 1$, but the translation $x \mapsto x + 1$ has no fixed point. The converse fails: $L(f) = 0$ does not prevent fixed points, as the identity of the circle shows.

Going deeper

For a student

The Lefschetz number depends only on the homotopy class of $f$, so the theorem gives fixed points for all maps in a homotopy class at once. It contains the Brouwer fixed-point theorem, since a closed ball is contractible. A refined form counts fixed points with indices: for a map with isolated fixed points, the sum of their fixed-point indices equals $L(f)$. In dynamics, applying it to the iterates $f^n$ gives periodic points, and the numbers $L(f^n)$ enter the Lefschetz zeta function.