Equivariant Lefschetz maps for simplicial complexes and smooth manifolds

Abstract

Let X be a locally compact space with a continuous proper action of a locally compact group G. Assuming that X satisfies a certain kind of duality in equivariant bivariant Kasparov theory, we can enrich the classical construction of Lefschetz numbers to equivariant K-homology classes. We compute the Lefschetz invariants for self-maps of finite-dimensional simplicial complexes and of self-maps of smooth manifolds. The resulting invariants are independent of the extra structure used to compute them. Since smooth manifolds can be triangulated, we get two formulas for the same Lefschetz invariant in these cases. The resulting identity is closely related to the equivariant Lefschetz Fixed Point Theorem of Luck and Rosenberg.

Minor revisions, affecting some theorem numbers

References in corpus (2)

• Dualities in equivariant Kasparov theory
• The universal functorial equivariant Lefschetz invariant

Cited by in corpus (6)

• Bivariant K-theory via correspondences
• Dualities in equivariant Kasparov theory
• Positive Scalar Curvature and Poincare Duality for Proper Actions
• A functorial equivariant K-theory spectrum and an equivariant Lefschetz formula
• Localization techniques in circle-equivariant KK-theory
• Structure and K-theory of crossed products by proper actions