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.