 Lefschetz fixed-point theorem

If $L(f)\ne 0$, then $f$ has a fixed point. For a fixed-point-free map, sufficiently fine simplicial approximation has zero diagonal chain coefficients, hence zero Lefschetz number.

The Lefschetz Fixed-Point Theorem is a fundamental result in algebraic topology that provides a criterion for determining the existence of fixed points of continuous maps on topological spaces. It is particularly useful when dealing with maps between compact, connected, and oriented manifolds.