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Concept
Lefschetz fixed-point theorem
Summary
In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space to itself by means of traces of the induced mappings on the homology groups of . 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 to itself. Define the Lefschetz number of by
the alternating (finite) sum of the matrix traces of the linear maps induced by on , the singular homology groups of with rational coefficients.
A simple version of the Lefschetz fixed-point theorem states: if
then has at least one fixed point, i.e., there exists at least one in such that . 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 has a fixed point as well.
Note however that the converse is not true in general: may be zero even if has fixed points, as is the case for the identity map on odd-dimensional spheres.
First, by applying the simplicial approximation theorem, one shows that if has no fixed points, then (possibly after subdividing ) is homotopic to a fixed-point-free simplicial map (i.e., it sends each simplex to a different simplex). This means that the diagonal values of the matrices of the linear maps induced on the simplicial chain complex of must be all be zero. Then one notes that, in general, the Lefschetz number can also be computed using the alternating sum of the matrix traces of the aforementioned linear maps (this is true for almost exactly the same reason that the Euler characteristic has a definition in terms of homology groups; see below for the relation to the Euler characteristic).
Official source
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