Formal Statement
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.
A stronger form of the theorem, also known as the Lefschetz-Hopf theorem, states that, if f has only finitely many fixed points, then
where Fix(f) is the set of fixed points of f, and i(f,x) denotes the index of the fixed point x.
Read more about this topic: Lefschetz Fixed-point Theorem
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