  We introduce homological and homotopical $r$-syzygies of Mori fibre spaces as
a generalization of Sarkisov links and relations of Sarkisov links. For any
proper morphism $Y/R$, we construct a contractible (if not empty) CW complex
such that there is a 1-1 correspondence between its cells and the central
models of $Y/R$. We derive from this CW complex a long exact sequence and a
spectral sequence converging to the (co)homology of the relative birational
automorphism group of $Y/R$. As an application, we compute the second group
homology of the second Cremona group.
