Let \(X\) be a finite simplicial complex, or a retract of one, and let \(f\colon X\to X\) be a continuous map. The Lefschetz number of \(f\) is \[ \tau(f) \coloneqq \sum_{n\geq 0} (-1)^n \tr\qty{ f_*\colon H_n(X;\QQ) \to H_n(X;\QQ) } .\]
Since \(\tr\qty{\id_{H_n(X;\QQ)}} = \dim_\QQ H_n(X;\QQ)\), the identity has \(\tau(\id_X) = \chi(X)\), the Euler characteristic. By the Lefschetz fixed point theorem, if \(\tau(f)\neq 0\), then \(f\) has a fixed point.
[@Hat02].