K-duality for stratified pseudomanifolds

Claire Debord (1) , Jean-Marie Lescure (1)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal

This paper is devoted to the study of Poincaré duality in K-theory for general stratified pseudomanifolds. We review the axiomatic definition of a smooth stratification $\fS$ of a topological space $X$ and we define a groupoid $T^{\fS}X$, called the $\fS$-tangent space. This groupoid is made of different pieces encoding the tangent spaces of the strata, and these pieces are glued into the smooth noncommutative groupoid $T^{\fS}X$ using the familiar procedure introduced by A. Connes for the tangent groupoid of a manifold. The main result is that $C^{*}(T^{\fS}X)$ is Poincaré dual to $C(X)$, in other words, the $\fS$-tangent space plays the role in $K$-theory of a tangent space for $X$.

Mots clés: Stratified spaces; pseudomanifolds; Lie groupoids; bivariant K-theory; index theory

Domaines: Algèbres d'opérateurs [math.OA]; K-théorie et homologie [math.KT]

Claire Debord, Jean-Marie Lescure. K-duality for stratified pseudomanifolds. 2007. ⟨hal-00214218v4⟩