Abstract
Let G be countable group and M be a proper cocompact even-dimensional G-manifold with orbifold quotient M M. Let D be a G-invariant Dirac operator on M. It induces an equivariant K-homology class [ D]. KG 0 ( M) and an orbifold Dirac operator D D on M M. Composing the assembly map K-0(G) (M) -> K-0(C*(G)) with the homomorphism K-0(C*(G)) -> Z given by the representation C*(G) -> C of the maximal ...
Abstract
Let G be countable group and M be a proper cocompact even-dimensional G-manifold with orbifold quotient M M. Let D be a G-invariant Dirac operator on M. It induces an equivariant K-homology class [ D]. KG 0 ( M) and an orbifold Dirac operator D D on M M. Composing the assembly map K-0(G) (M) -> K-0(C*(G)) with the homomorphism K-0(C*(G)) -> Z given by the representation C*(G) -> C of the maximal group C*-algebra induced from the trivial representation of G we define index([D]). Z. In the second section of the paper we show that index(D) D) = index([D]) and obtain explicit formulas for this integer. In the third section we review the decomposition of KG(0) (M) in terms of the contributions of fixed point sets of finite cyclic subgroups of G obtained by W. Luck. 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.
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