Combining Kasparov's generalization of a theorem of Voiculescu and Cuntz's description of KK-theory in terms of quasihomomorphisms (sections one and two), we give a simple construction of the Kasparov product (section three). This construction will be generalized in [Gre] to give a version of the product for so-called locally convex Kasparov modules over locally convex algebras in order to treat products of certain universal cycles.

We give a classification (up to smooth homotopy) of finitely summable Fredholm representations (Fredholm modules) over higher rank groups and lattices. Our results are a direct consequence of work of Bader, Furman, Gelander and Monod on generalizations of Kazhdan's property T for locally compact groups.

In [7], a notion of noncommutative tangent space is associated with a conical pseudomanifold and Poincaré duality in K-theory is proved between this space and the pseudomanifold. The present paper continues this line of work. We show that an appropriate presentation of the notion of symbol on a manifold generalizes right away to conical pseudomanifolds and that it enables us to interpret Poincaré duality in the singular setting as a noncommutative symbol map.