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 J. Deng, G. Tian, Z. Xie and G. Yu 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.
Wang et al. (Fri,) studied this question.