Lefschetz fixed-point theorem
In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named after Solomon Lefschetz, who first stated it in 1926.
The counting is subject to an imputed multiplicity at a fixed point called the fixed point index. 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 of the theorem, let
be a continuous map from a compact triangulable space X to itself. Define the Lefschetz number Λf of f by
the alternating (finite) sum of the matrix traces of the linear maps induced by f on the Hk(X,Q), the singular homology of X with rational coefficients.
A simple version of the Lefschetz fixed-point theorem states: if
then f has at least one fixed point, i.e. there exists at least one x in X such that f(x) = x. In fact, since the Lefschetz number has been defined at the homology level, the conclusion can be extended to say that any map homotopic to f has a fixed point as well.
Note however that the converse is not true in general: Λf may be zero even if f has fixed points.
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