Solomon Lefschetz wanted to count the fixed point set of a continuous map.
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
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.
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 Brower fixed point theorem.
The existence of a Lefschetz formula holds more general in Weil cohomology theories (by definition) and hence notably in ∞-adic? étale cohomology. This fact serves to prove the Weil conjectures.
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The original article is
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 of schemes:
For algebraic stacks: