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\(K\)-duality for stratified pseudomanifolds. (English) Zbl 1159.19303
Summary: This paper continues our project started in [C. Debord and J.-M. Lescure, J. Funct. Anal. 219, No. 1, 109–133 (2005; Zbl 1065.58014)] 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 \(C^{\ast}(T^SX)\) is Poincaré dual to \(C(X)\), in other words, the \(S\)-tangent space plays the role in \(K\)-theory of a tangent space for \(X\).

MSC:
19K35 Kasparov theory (\(KK\)-theory)
46L80 \(K\)-theory and operator algebras (including cyclic theory)
57N80 Stratifications in topological manifolds
58B34 Noncommutative geometry (à la Connes)
58H05 Pseudogroups and differentiable groupoids
19K33 Ext and \(K\)-homology
19K56 Index theory
57P99 Generalized manifolds
58A35 Stratified sets
58J20 Index theory and related fixed-point theorems on manifolds
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