EquivariantL-theory I

Abstract

Equivariant algebraic K-theory decomposes as a sum of ordinary algebraic Ktheory of group rings (see [4, 5, 7, 12]). This follows for K t because the determinant of a upper triangular matrix is the product of the entries on the diagonal. For equivariant L-theory the situation is more complicated. We do obtain a set of exact orbit sequences similar to the neighbouring family sequences obtained by Connor and Floyd in [2]. But these exact sequences do not always split in a way analogous to the splitting of equivariant K-theory. There are easy counterexamples for G=Z/2 . However, if the transformation group has odd order, then the equivariant L-groups do in fact decompose in the expected fashion, cf. Theorem 2.11 below. We work in this paper in the smooth and locally linear PL-category simultaneously. A consequence of the existence of the exact orbit sequence is that the equivariant L-groups are equal for these two manifold categories. The paper is founded upon the definition of equivariant L-theory given in [9]. We refer the reader to that paper for the somewhat cumbersome definitions. A reference (I.?. ?) always refers to the first part [9]. It is our hope that the present definitions of equivariant L-groups and the calculational techniques presented here will make further calculations possible. It seems to us to be of some interest to evaluate equivariant L-groups for some of the standard 2-groups, for example, and to determine the equivariant Rothenberg sequence.