Lefschetz fixed point theorem

Let \(X\) be a finite simplicial complex, or a retract of one, and let \(f\colon X\to X\) be continuous with Lefschetz number \[ \tau(f) = \sum_{k \geq 0} (-1)^k \tr\qty{f_*\colon H_k(X; \QQ) \to H_k(X; \QQ)} .\] If \(\tau(f) \neq 0\), then \(f\) has a fixed point [@Hat02].