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Lefschetz trace formula

Contents

Motivation

Solomon Lefschetz wanted to count the fixed point set of a continuous map.

Lefschetz number

Fix a ground field kk. Given a continuous map f:XXf \colon X\to X of topological space, its Lefschetz number Λ(X,f)\Lambda(X,f) is the alternating sum of the traces

i(1) iTr(H i(f):H i(X,k)H i(X,k), \sum_i (-1)^i Tr (H^i(f) \colon H^i(X,k)\to H^i(X,k) \,,

of the endomorphisms of the ordinary cohomology groups with coefficients in the ground field kk.

One sometimes also talks of the Lefschetz number of the induced endomorphism of the chain/cochain complexes, see algebraic Lefschetz formula.

For f=idf = id the identity map, the Lefschetz trace reduces to the Euler characteristic.

Lefschetz fixed point theorem

Statement

The Lefschetz fixed point theorem says that if the Lefschetz number is non-zero, then ff has at least one fixed point.

The existence of a Lefschetz formula holds more general in Weil cohomology theories (by definition) and hence notably in ℓ-adic étale cohomology. This fact serves to prove the Weil conjectures.

Proof

(…)

follows from existence of

(Milne, section 25)

(…)

References

For ordinary cohomology

The original article is

  • Solomon Lefschetz, On the fixed point formula, Ann. of Math. (2), 38 (1937) 819–822

Reviews include

See also

  • Minhyong Kim, A Lefschetz trace formula for equivariant cohomology, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 28 no. 6 (1995), p. 669-688, numdam, MR97d:55012

  • Atiyah, Bott, … (cf. Atiyah-Bott fixed point formula)

For étale cohomology

For étale cohomology of schemes:

For algebraic stacks:

  • Kai Behrend, The Lefschetz trace formula for algebraic stacks, Invent. Math. 112, 1 (1993), 127-149, doi

Revised on November 22, 2013 05:48:20 by Urs Schreiber (82.169.114.243)