$K$–duality for stratified pseudomanifolds

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 $T^{S}X$, 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 $X$.