Lefschetz number

Summary

Let $f$ be a self-map of a finite CW complex or compact polyhedron. Its Lefschetz number is the alternating sum of the traces of the maps that $f$ induces on rational homology. It depends only on the homotopy class of $f$, and for the identity map it is the Euler characteristic. The Lefschetz fixed-point theorem says that if it is not zero, then $f$ has a fixed point.