Orbifold index and equivariant K-homology

Abstract

Let G be countable group and M be a proper cocompact even-dimensional G-manifold with orbifold quotient \(\bar{M}\) . Let D be a G-invariant Dirac operator on M. It induces an equivariant K-homology class \([D] \in K^G_0(M)\) and an orbifold Dirac operator \(\bar{D}\) on \(\bar{M}\) . Composing the assembly map \(K^G_0(M) \rightarrow K_0(C*(G))\) with the homomorphism \(K_0(C^*(G)) \rightarrow {\mathbb{Z}}\) given by the representation \(C^*(G) \rightarrow {\mathbb{C}}\) of the maximal group C *-algebra induced from the trivial representation of G we define index\(([D]) \in {\mathbb{Z}}\) . In the second section of the paper we show that index\((\bar{D})\) = index([D]) and obtain explicit formulas for this integer. In the third section we review the decomposition of \(K_0^G(M)\) in terms of the contributions of fixed point sets of finite cyclic subgroups of G obtained by W. Lück. In particular, the class [D] decomposes in this way. In the last section we derive an explicit formula for the contribution to [D] associated to a finite cyclic subgroup of G.