The Kasparov product on submersions of open manifolds

We study the Kasparov product on (possibly non-compact and incomplete) Riemannian manifolds. Specifically, we show on a submersion of Riemannian manifolds that the tensor sum of a regular vertically elliptic operator on the total space and an elliptic operator on the base space represents the Kasparov product of the corresponding classes in [Formula: see text]-theory. This construction works in general for symmetric operators (i.e. without assuming self-adjointness), and extends known results for submersions with compact fibers. The assumption of regularity for the vertically elliptic operator is not always satisfied, but depends on the topology and geometry of the submersion, and we give explicit examples of non-regular operators. We apply our main result to obtain a factorization in unbounded [Formula: see text]-theory of the fundamental class of a Riemannian submersion, as a Kasparov product of the shriek map of the submersion and the fundamental class of the base manifold.

On the L2-Poincaré duality for incomplete Riemannian manifolds: A general construction with applications

Let [Formula: see text] be an open, oriented and incomplete Riemannian manifold of dimension [Formula: see text]. Under some general conditions we show the existence of a Hilbert complex [Formula: see text] such that its cohomology groups, labeled with [Formula: see text], satisfy the following properties: [Formula: see text] [Formula: see text] (Poincaré duality holds) There exists a well-defined and nondegenerate pairing: [Formula: see text] If [Formula: see text] is a Fredholm complex, then every closed extension of the de Rham complex [Formula: see text] is a Fredholm complex and, for each [Formula: see text], the quotient [Formula: see text] is a finite dimensional vector space.