The Novikov conjecture for low degree cohomology classes

We outline a twisted analogue of the Mishchenko-Kasparov approach to prove the Novikov conjecture on the homotopy invariance of the higher signatures. Using our approach, we give a new and simple proof of the homotopy invariance of the higher signatures associated to all cohomology classes of the classifying space that belong to the subring of the cohomology ring of the classifying space that is generated by cohomology classes of degree less than or equal to 2, a result that was first established by Connes and Gromov and Moscovici using other methods. A key new ingredient is the construction of a tautological C*(r) (Gamma,sigma)-bundle and connection, which can be used to construct a C*(r) (Gamma,sigma)-index that lies in the Grothendieck group of C*(r)(Gamma,sigma), where sigma is a multiplier on the discrete group Gamma corresponding to a degree 2 cohomology class. We also utilise a main result of Hilsum and Skandalis to establish our theorem.