# Contents

## Motivation

Solomon Lefschetz wanted to count the fixed point set of a

## Lefschetz number

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

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

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

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

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

## Lefschetz fixed point theorem

### Statement

The Lefschetz fixed point theorem says that if $X$ is a compact polyhedron and if the Lefschetz number is non-zero, then $f$ has at least one fixed point. In particular, if $X$ is a contractible compact polyhedron, then every $f \colon X\to X$ has a fixed point, so the theorem is a vast generalization of Brouwer's fixed point theorem.

The existence of a Lefschetz formula holds more generally 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

(…)

## 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

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