 Lefschetz fixed-point theorem

Let $X$ be the geometric realization of a finite simplicial complex, or a retract of such a space, and let $f:X\to X$ be continuous. If its Lefschetz number $L(f)$ is nonzero, then $f$ has a fixed point. This includes self-maps of compact manifolds.^[ref]

For a map without fixed points on a finite polyhedron, compactness gives a positive separation between $x$ and $f(x)$. After sufficiently fine simplicial subdivision, a simplicial approximation sends no simplex back onto itself. Its diagonal chain map coefficients vanish, so its alternating trace, and hence $L(f)$, is zero. A retraction reduces the more general stated case to this one.

For example, every continuous self-map of an even-dimensional sphere with mapping degree different from $-1$ has a fixed point, since $L(f)=1+\mathrm{deg}f$. The converse fails: $L(f)=0$ permits fixed points, as the identity map of a circle shows.

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.