Equivariant Lefschetz maps for simplicial complexes and smooth manifolds

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.

💡 Research Summary

The paper develops an equivariant extension of the classical Lefschetz number by embedding it into Kasparov’s bivariant K‑theory. The setting is a locally compact Hausdorff space (X) equipped with a continuous proper action of a locally compact group (G). The authors assume that (X) possesses an equivariant Poincaré‑duality element (D\in KK^{G}(C_{0}(X),\mathbb{C})); this duality is equivalent to saying that (X) is (G)-oriented in the sense of equivariant K‑theory. Under this hypothesis, any (G)-equivariant self‑map (f:X\to X) determines a class (