A Spectral sequence for polynomially bounded cohomology

TL;DR

This paper develops a spectral sequence for polynomial cohomology in group extensions, linking properties of subgroups to the entire group and implications for the Novikov Conjecture.

Contribution

It introduces a Lyndon-Hochschild-Serre spectral sequence analogue for polynomial cohomology in group extensions, connecting subgroup properties to the whole group.

Findings

Spectral sequence converges to polynomial cohomology of G.

Polynomial cohomology of G relates to that of Q and H under certain conditions.

Groups with properties of Q and H satisfy the Novikov Conjecture.

Abstract

We construct an analogue of the Lyndon-Hochschild-Serre spectral sequence in the context of polynomial cohomology, for group extensions. If G is an extension of Q by H, then the spectral sequence converges to the polynomial cohomology of G. For the polynomial extensions of Noskov with normal subgroup isocohomological, the E_2 term is the polynomial cohomology of Q with coefficients in the polynomial cohomology of H. When both Q and H are isocohomological G must be as well. By referencing results of Connes-Moscovici and Noskov, if Q and H are both isocohomological and have the Rapid Decay property of Jolissaint, then G satisfies the Novikov Conjecture.