  We consider a pairing producing various cyclic Hochschild cocycles, which led
Alain Connes to cyclic cohomology. We are interested in geometrical meaning and
homological properties of this pairing. We define a non-trivial pairing between
the homology of a Lie-Rinehart (super-)algebra with coefficients in some
partial traces and relative periodic cyclic homology. This pairing generalizes
the index formula for summable Fredholm modules, the Connes-Kubo formula for
the Hall conductivity and the formula computing the K0-group of a smooth
noncommutative torus. It also produces new homological invariants of proper
maps contracting each orbit contained in a closed invariant subset in a
manifold acted on smoothly by a connected Lie group. Finally we compare it with
the characteristic map for the Hopf-cyclic cohomology.
