K-duality for pseudomanifolds with isolated singularities

We associate to a pseudomanifold $X$ with isolated singularities a differentiable groupoid $G$ which plays the role of the tangent space of $X$. We construct a Dirac element $D$ and a Dual Dirac element $\lambda$ such that $D$ and $\lambda$ induce a Poincar{\'e} duality in $K$-theory between the $C^*$-algebras $C(X)$ and $C^*(G)$.