The computation of odd dimensional projective surgery groups of finite groups

Abstract

§0. INTRODUCTION THiS PAPER computes the odd dimensional, projective, surgery groups LPn+I(~) for all finite groups 7r. The computat ion is made as follows. We reduce by Dress induction to the case 1r is 2-hyperelementary and here, we express LPn+l(Tr) as a direct sum of class groups C(X) where X ranges over the irreducible characters of ~r. Several precise numerical calculations are deduced from the general computation. One such calculation is the following: If the Schur index of any irreducible character of a 2-hyperelementary subgroup of ,r is trivial then L1e(Tr) -0; this is the case, for example, if either zr itself or, more generally, every 2-hyperelementary subgroup of is a direct product of abelian and dihedral groups. Another precise calculation is the following: If the 2-hyperelementary subgroups of ~r are abelian then L 3 P ( T T ) (Z[2Z) r°-I where r0 is the number of real, irreducible characters of ~-. The latter calculation and a special case of the former have appeared already in Bak [4]. It is worth mentioning that the even dimensional, projective, surgery groups L~',(~') of finite groups 7r are computed in the following papers; in the case that the 2-hyperelementary subgroups of ~ are abelian, they have been computed already in Bak[4 ,7] and Bak-Schar lau[9] and in the general case, they are computed in Kolster [21] and Kolster [21.1]. We describe now the organization and results of the paper. In §l, we recall briefly the required notation from the Ktheory of quadratic forms; in particular, if A is an involution invariant subring of a semisimple ring B with involution and if A = _+ l then fWo~(A) denotes the Witt group of A-forms (equivalently nonsingular min-quadratic modules) on finitely generated, project ive A-modules which become free of even rank over B. We then cite an exact sequence of Witt groups due to Pardon and Ranicki. In §2, we compute the cokernel of the canonical homorphism