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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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