Relative bounded cohomology

In this paper, we provide the algebraic foundations to the theory of relative bounded cohomology. In particular, we prove that the relative bounded cohomology of spaces depends only on their fundamental groups. We also prove Gromovʹs Equivalence theorem and Relative mapping theorem for relative bounded cohomology, stated by Gromov without proofs in [M. Gromov, Publ. Math. IHES 56 (1982) 5–100].