Generalized Lefschetz numbers for equivariant maps

Abstract

In this paper we discuss a kind of Lefschetz number defined for a generalized multiplicative -equivariant cohomology theory ∗ = { } where is a compact Lie group, and ∈ I runs along some at most countable set of indices I (cf. [22, 3]). For every pair ⊂ of finite -CW-complexes the cohomology algebra ∗ ( ) has the structure of an ∗ (pt)-module and, consequently, also of an 0 (pt)-module, given by the multiplicative structure. Assume that ∗ ( ) is either (a) a finitely generated projective ∗ (pt)-module, or (b) a finitely generated projective 0 (pt)-module respectively. Let : ( ) −→ ( ) be a -equivariant map. Under assumption (a) a generalized trace, tr ∗, of the induced map ∗ : ∗ ( ) −→ ∗ ( ) is well defined (cf. [23, 4]) and will be called the full generalized Lefschetz number and denoted by