Lefschetz fixed-point theorem

The Lefschetz fixed-point theorem is a mathematical tool that predicts whether a function acting on a shape (like a surface or space) has to have at least one point that it maps to itself. It uses an algebraic calculation called the Lefschetz number, derived from how the function transforms different parts of the shape. If this number isn't zero, the theorem guarantees there's at least one fixed point—a point unchanged by the function. It's a powerful way to connect abstract algebraic properties with the concrete existence of fixed points in various mathematical contexts.