K-theory of relative group C∗-algebras and extensions of coarsely embeddable groups

The relative Novikov conjecture for manifolds with boundary states that the relative higher signatures of such manifolds are invariant under orientation-preserving homotopy equivalences of pairs. It was proved to be true by Deng et al. when the fundamental group of the whole manifold admits a coarse embedding into Hilbert space, and the fundamental group of the boundary is a-T-menable. In this paper, we study a case in which the fundamental group of the whole manifold does not necessarily admit a coarse embedding into a Hilbert space. More precisely, we prove that if [Formula: see text] and [Formula: see text] are orientation-preserving homotopy equivalent, [Formula: see text] is a normal subgroup of [Formula: see text], and both [Formula: see text] and [Formula: see text] admit coarse embeddings into Hilbert space, then the relative higher signatures of [Formula: see text] and [Formula: see text] are equal.