On the Poincaré isomorphism in $K$-theory on manifolds with edges

In this paper, the Poincaré isomorphism in $K$-theory on manifolds with edges is constructed. It is shown that the Poincaré isomorphism can be naturally constructed in terms of noncommutative geometry. More precisely, we obtain a correspondence between a manifold with edges and a noncommutative algebra and establish an isomorphism between the $K$-group of this algebra and the $K$-homology group of the manifold with edges, which is considered as a compact topological space.