Algebraic X - Theory of Poly - ( Finite or Cylic ) Groups

The K-theory of the title is described in terms of the Ktheory of finite subgroups, as generalized sheaf homology of a quotient space. A corollary is that if G is torsion-free, then the Whitehead groups Wh t(ZG) vanish for all i. 1. The main result. Suppose that G is a poly-(finite or cyclic) group. Then there is a virtually connected and solvable Lie group L that contains G as a discrete cocompact subgroup. ("Virtually" means a subgroup of finite index has the indicated property.) This follows from results of Auslander and Johnson, as observed in [5]. Let K be a maximal compact subgroup of L. Then G acts (on the right) on the contractible manifold K\L. The action may not be free; for y e {K\L) the isotropy subgroup is Gy = {yGy~ ) D K. These isotropy subgroups are finite since they are discrete in the compact group K. Consider the quotient K\L/G. Let [y] denote a point with preimage y in K\L. Then the isotropy subgroup Gy is determined by [y] up to conjugacy. Gy therefore defines a "cosheaf up to conjugacy" of groups over K\L/G. If R is a ring we can apply the algebraic K-theory functor K(i2[*]) to this to get a cosheaf of spectra over K\L/G. Homology groups with coefficients in a spectral cosheaf are defined [12] and, in this case, denoted by Hl(K\L/G\K.(RGy)). Homology is discussed further in §2. The K-theory spectrum used here is the nonconnective one; the lower homotopy groups of the spectrum are Bass's groups K-3;. Also, maps of K(RG) induced by conjugation of G are essentially canonically homotopic to the identity. Therefore the uncertainty of definition of the Gy vanishes at the spectrum level. 1.1 THEOREM. Suppose G is a discrete cocompact subgroup of a Lie group L that is virtually connected and solvable and has maximal compact subgroup K. Suppose R is a subring of the rationals in which the order of the torsion of G is invertible. Then the natural homomorphism Ht(K\L/G;K{RGy)) -» Kt(RG) is an isomorphism for all i. Before discussing the proof we give corollaries and references. First, if G is virtually abelian Yamasaki [18] has proved the analog of 1.1 for the surgery groups of G. Received by the editors October 29, 1984. 1980 Mathematics Subject Classification. Primary 16A54, 18F25, 22E40. ©1985 American Mathematical Society 0273-0979/85 $1.00 + $.25 per page