# Contents

## Motivation

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

## Lefschetz number

Given a continuous map $f:X\to X$, its Lefschetz number $\Lambda \left(X,f\right)$ is the alternating sum of the traces

$\sum _{i}\left(-1{\right)}^{i}\mathrm{Tr}\left({H}^{i}\left(f\right):{H}^{i}\left(X,k\right)\to {H}^{i}\left(X,k\right)\phantom{\rule{thinmathspace}{0ex}},$\sum_i (-1)^i Tr (H^i(f):H^i(X,k)\to H^i(X,k) \,,

of cohomology with coefficients in the (in advance fixed) 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 $f=\mathrm{id}$ the identity map, the Lefschetz trace reduces to the Euler characteristic.

## Lefschetz trace formula

(…) See for instance the eom article.

## References

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

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

• 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)

Revised on March 13, 2012 18:35:22 by Urs Schreiber (82.169.65.155)