nLab
Lefschetz trace formula

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 $\Lambda (X,f)$ is the alternating sum of the traces

$$\sum _i(-1{)}^i\mathrm{Tr}(H^i(f):H^i(X,k)\to H^i(X,k),$$\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.

Equivariant case

…

Related concepts

• Reidemeister trace

References

• Online Springer Enc. of Math. (eom): Lefschetz number (Rudyak), Lefschetz formula (Iskovskikh)

• 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)