Witt Spaces: A Geometric Cycle Theory for KO-Homology at Odd Primes

Abstract

Introduction. This paper studies a class of stratified piecewiselinear pseudomanifolds, which we call Witt spaces, characterized by natural local intersection homology conditions [11]. We compute the cobordism groups by introducing an invariant taking values in the Witt group of symmetric bilinear forms over the rationals, W(Q). These pseudomanifolds solve a problem posed by D. Sullivan [25]: to construct a class of P.L. cycles with signature which represent the connected version of KO homology at odd primes, ko* 0D Z[1/2]. The result was one of the first applications of the intersection homology theory of Goresky and MacPherson [11]. Specifically, the rational intersection homology groups of Witt spaces satisfy a Poincare duality theorem. To each Witt space X, we associate a P.L. invariant w(X) with values in W(Q). This invariant generalizes the signature of and satisfies cobordism invariance, additivity, and a product formula. Adapting classical surgery (spherical modification) to this setting, we prove that w(X) determines the cobordism class of X and obtain an explicit description of the Witt cobordism groups. The only nontrivial groups occur in dimensions 4k, and for k > 0 they are W(Q). Sullivan [25,27] has shown that a class of P.L. pseudomanifolds equipped with such an invariant possesses canonical orientations in ko* (0 Z[1/2]. The orientations induce a natural transformation it of homology theories, from Witt space bordism to ko* 0)Z[1/2]. The structure of W(Q) is known [14], and we conclude that it is an equivalence at odd primes. In this paper, we study the cobordism theory of a class of stratified P.L. pseudomanifolds by means of the recent intersection homology theory of Goresky and MacPherson [11]. Goresky and MacPherson were interested in finding a class of spaces with cobordism invariant signature for the purpose of extending the Hirzebruch L-class to the setting of manifolds with