Equivariant Lefschetz maps for simplicial complexes and smooth manifolds (Q842374)

Let \(X\) be a locally compact Hausdorff space, let \(G\) be a locally compact group, and suppose that \(G\) acts properly and continuously on \(X\). A \(G\)-equivariant abstract dual for \(X\) is a \(G\)-\(C^*\)-algebra \(\mathcal{P}\) and a class \(\Theta\) in \(\text{RKK}^G_n(X; \mathbb{C}, \mathcal{P})\) for some \(n\) such that Kasparov product with \(\Theta\) induces isomorphisms \(\text{KK}^G_*(\mathcal{P}\otimes A, B) \cong \text{RKK}^G_{*+n}(X; A, B)\) for all \(G\)-\(C^*\)-algebras \(A\) and \(B\). Given an abstract dual \((\mathcal{P}, \Theta)\), the authors define an equivariant Lefschetz map \(\text{Lef}: \text{RKK}^G_*(X, C_0(X), \mathbb{C}) \rightarrow \text{KK}^G_*(C_0(X), \mathbb{C})\). There is a canonical homomorphism from \(\text{KK}^G_*(C_0(X), C_0(X))\) to \(\text{RKK}^G_*(X, C_0(X), \mathbb{C})\), and thus every continuous function \(\psi\) from \(X\) to itself determines an element of \(\text{RKK}^G_0(X, C_0(X), \mathbb{C})\) via this canonical map. By then applying the Lefschetz map, one obtains an element \(\text{Lef}(\psi)\) of \(\text{KK}^G_0(C_0(X), \mathbb{C})\); this element is independent of the abstract dual chosen. The authors compute \(\text{Lef}(\psi)\) in two important cases: when \(X\) is a smooth manifold and when \(X\) is a simplicial complex. Because every smooth manifold is triangulable, the author have two formulas for \(\text{Lef}(\psi)\) when \(X\) is a smooth manifold and \(\psi\) is a smooth map. The equality of these two formulas can be viewed as a Lefschetz fixed point theorem in \(\text{KK}^G\).