K–duality for stratified pseudomanifolds

Debord, Claire¹ ; Lescure, Jean-Marie¹

¹ Laboratoire de Mathématiques, Université Blaise Pascal, Complexe universitaire dse Cézeaux, 24 Av. des Landais, 63177 Aubière cedex, France

Geometry & topology, Tome 13 (2009) no. 1, pp. 49-86.

Résumé

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 $X$.

Détail

DOI : 10.2140/gt.2009.13.49

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

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