On syzygies of Mori fibre spaces

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 $Y\mathrm{/}R$, 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 $Y\mathrm{/}R$. 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 $Y\mathrm{/}R$. As an application, we compute the second group homology of the second Cremona group.