Lefschetz fixed-point theorem

Statement

If the Lefschetz number of a map from a compact polyhedron (viz a compact space that is also a polyhedron) to itself is nonzero, then the map has a fixed point.

Corollaries

• Any contractible compact polyhedron has the fixed-point property. In particular, every disc has the fixed-point property, which is Brouwer fixed-point theorem
• Rationally acyclic compact polyhedron has fixed-point property
• Euler characteristic of compact connected nontrivial Lie group is zero: Euler characteristic of any nontrivial compact connected Lie group is zero
• Self-map of sphere of degree greater than one has a fixed point: Any map from a sphere to itself of degree greater than 1 must have a fixed point