K-duality for stratified pseudomanifolds

TL;DR

This paper establishes Poincaré duality in K-theory for stratified pseudomanifolds by constructing an $S$-tangent groupoid that generalizes tangent spaces to stratified spaces, showing its $C^*$-algebra is dual to the space.

Contribution

It introduces the $S$-tangent space groupoid for stratified pseudomanifolds and proves its $C^*$-algebra is Poincaré dual to the space, extending tangent space concepts.

Findings

$C^{*}(T^{S}X)$ is Poincaré dual to $C(X)$

The $S$-tangent space encodes tangent data of strata

Generalizes tangent groupoid to stratified pseudomanifolds

Abstract

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 $\fS$ 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 Poincar\'e dual to $C(X)$, in other words, the $\fS$-tangent space plays the role in $K$-theory of a tangent space for $X$.