Lefschetz fixed-point theorem

In mathematics, the Lefschetz fixed-point theorem counts the number of fixed points of a mapping from a topological space X to itself (subject to some mild conditions on X), by means of traces of the induced mappings on the homology groups of X. The counting is subject to some imputed multiplicity at a fixed point. A weak version of the theorem is enough to show that a mapping without any fixed point must have rather special topological properties (like a rotation of a circle).

For a formal statement, let

$f:X \rightarrow X$

be a continuous map from a compact triangulable space X to itself. A point x of X is a fixed point of f if f(x)=x. Denote the Lefschetz number of f by

$\Lambda_f.\,$

By definition this is

$\sum(-1)^k\mathrm{Tr}(f_*|H_k(X,Q))$ ,

the alternating (finite) sum of the matrix traces of the linear maps induced by f on the homology of X, with rational number coefficients.

Then the Lefschetz fixed-point theorem states that if

$\Lambda_f \neq 0$ ,

then f has a fixed point. In fact, since the Lefschetz number has been defined at the homology level, our conclusion can be extended to say that any map homotopic to f has a fixed point.

Categories: Topology | Theorems