Lefschetz fixed-point theorem
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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
- <math>f:X \rightarrow X<math>
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
- <math>\Lambda_f.\,<math>
By definition this is
- <math>\sum(-1)^k\mathrm{Tr}(f_*|H_k(X,Q))<math>,
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
- <math>\Lambda_f \neq 0<math>,
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.
Historical Context
Lefschetz presented his fixed point theorem in his 1926 paper Intersections and Transformations of Complexes and Manifolds in volume 28 of the Transactions of the American Mathematical Society. Lefschetz's focus was not on fixed points of mappings, but rather on what are now called coincidence points of mappings.
Given two maps f and g from a manifold X to a manifold Y, the Lefschetz coincidence number of f and g is defined as
- <math>\Lambda_{f,g} = \sum (-1)^k \mathrm{Tr}( D_X \circ g^* \circ D_Y^{-1} \circ f_*) <math>,
where f* is as above, g* is the mapping induced by g on the cohomology groups with rational number coefficients, and DX and DY are the Poincaré duality isomorphisms for X and Y, respectively.
Lefschetz proves that if the coincidence number is nonzero, then f and g have a coincidence point. He notes in his paper that letting X=Y and letting g be the identity map gives a simpler result, which we now know as the fixed point theorem.