K-duality for stratified pseudomanifolds

This paper is devoted to the study of Poincar'e duality in K-theory for general stratified pseudomanifolds. We review the axiomatic definition of a smooth stratification $\text{fS}$ of a topological space $X$ and we define a groupoid $T^{\text{fS}}X$, called the $\text{fS}$-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 $T^{\text{fS}}X$ using the familiar procedure introduced by A. Connes for the tangent groupoid of a manifold. The main result is that $C^{\ast }(T^{\text{fS}}X)$ is Poincar'e dual to $C(X)$, in other words, the $\text{fS}$-tangent space plays the role in $K$-theory of a tangent space for $X$.