#### Volume 13, issue 1 (2009)

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$K$–duality for stratified pseudomanifolds

### Claire Debord and Jean-Marie Lescure

Geometry & Topology 13 (2009) 49–86
##### 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 ${T}^{S}X$, 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 ${T}^{S}X$ using the familiar procedure introduced by Connes for the tangent groupoid of a manifold. The main result is that ${C}^{\ast }\left({T}^{S}X\right)$ is Poincaré dual to $C\left(X\right)$, 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 2000
Primary: 58B34, 46L80, 19K35, 58H05, 57N80
Secondary: 19K33, 19K56, 58A35, 57P99
##### Publication
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
 Claire Debord Laboratoire de Mathématiques Université Blaise Pascal Complexe universitaire dse Cézeaux 24 Av. des Landais 63177 Aubière cedex France http://math.univ-bpclermont.fr/~debord/ Jean-Marie Lescure Laboratoire de Mathématiques Université Blaise Pascal Complexe universitaire dse Cézeaux 24 Av. des Landais 63177 Aubière cedex France http://math.univ-bpclermont.fr/~lescure/