Volume 13, issue 1 (2009)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
$K$–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 TSX, 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 TSX using the familiar procedure introduced by Connes for the tangent groupoid of a manifold. The main result is that C(TSX) 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 2000
Primary: 58B34, 46L80, 19K35, 58H05, 57N80
Secondary: 19K33, 19K56, 58A35, 57P99
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
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/