Geometry & Topology 13 (2009) 49–86

DOI: 10.2140/gt.2009.13.49

Abstract

This paper continues our project started in [J. Funct. Anal. 219, 109–133] where Poincaré duality in K–theory was studied for singular manifolds with isolated conical singularities. Here, we extend the study and the results to general stratified pseudomanifolds. We review the axiomatic definition of a smooth stratification S of a topological space X and we define a groupoid T^SX, called the S–tangent space. This groupoid is made of different pieces encoding the tangent spaces of strata, and these pieces are glued into the smooth noncommutative groupoid T^SX using the familiar procedure introduced by Connes for the tangent groupoid of a manifold. The main result is that C^*(T^SX) is Poincaré dual to C(X), in other words, the S–tangent space plays the role in K–theory of a tangent space for X.

Keywords

singular manifolds, smooth groupoids, Kasparov bivariant K–theory, Poincaré duality

Mathematical Subject Classification

Primary: 19K35, 46L80, 57N80, 58B34, 58H05

Secondary: 19K33, 19K56, 57P99, 58A35

References

Publication

Received: 20 February 2008
Revised: 18 August 2008
Accepted: 4 July 2008
Preview posted: 22 October 2008
Published: 1 January 2009
Proposed: Steve Ferry
Seconded: Ralph Cohen, Wolfgang Lueck

Authors