Strong Relative Novikov Conjecture

Abstract

This dissertation can be said to consider Relative Strong Novikov Conjecture for a pair of countable discrete groups. The first part of the dissertation is about formulation of the relative Baum-Connes assembly map for a pair of countable discrete groups. Our goal is to extend the theory to relative case so that it becomes applicable to relative Novikov conjecture for manifold with boundary. Different from the classical case, we have to consider maximal group C ∗ -algebras since it is functorial in nature. In the second part of the dissertation, we study when the strong relative Novikov conjecture is true. Yu and Skandalis-Tu-Yu proved that if a group (viewed as metric spaces with respect to a word metric) admits a coarse embedding into a Hilbert space, then the strong Novikov conjecture is true. Suppose h : G → Γ is a group homomorphism. In the relative case, we will prove that if G is an a-T-menable group, Γ admits a coarse embedding into a Hilbert space, then the strong relative Novikov conjecture is true. Secondly, we will prove that if ker(h) is trival and Γ admits a coarse embedding into a Hilbert space, then the strong relative Novikov conjecture is true.