Lefschetz fixed-point theorem

The Lefschetz fixed-point theorem states that a continuous self-map of a suitable space has a fixed point when its Lefschetz number is nonzero.

Lefschetz number: An alternating sum of traces of a map's induced homology maps. Its nonvanishing is the theorem's criterion for guaranteeing a fixed point.
Singular homology: A homology theory built from continuous maps of simplices into a topological space. It supplies the homology groups commonly used to define the Lefschetz number.
Identity map: A function that maps every element of a set to itself. Its Lefschetz number is the Euler characteristic, and every point is fixed.
Nielsen fixed-point theory: A theory that classifies fixed points into homotopy-invariant classes and estimates their minimum number. It refines existence questions beyond the nonzero Lefschetz-number test.
Induced map: A map between algebraic invariants produced by a continuous map between topological spaces. The theorem computes traces from the self-map's induced maps on homology.
Trace (linear algebra): The sum of the diagonal entries of a square matrix, invariant under change of basis. The theorem sums traces of linear maps induced on homology.
Constant map: A function that sends every point of its domain to one fixed value. On a nonempty space, its constant value is a fixed point of the self-map.
Banach fixed-point theorem: A contraction on a complete metric space has a unique fixed point, reached by iteration. It uses metric contraction rather than homology to guarantee a unique fixed point.
Homology group: An algebraic structure measuring cycles and holes in a topological space. The Lefschetz number combines traces on the space's homology groups.
Continuous function: A function between topological spaces for which nearby inputs have nearby outputs in the topological sense. Continuity is required for the self-map to induce maps on homology.

Linked from 5 pages