(L) at (-1,0) [$L'_p$]{}; in [0,.8]{} [ in [0,1,2,3,4,6,7,8,9]{} [ (,) circle (.7mm);]{} (1,) – (2,); (2,) – (3,); (3,) – (4,); (6,) – (7,); (7,) – (8,); (8,) – (9,); (0,+ .05) – (1,+ .05); (0,- .05) – (1,-.05); ]{} (4,.8) – (6,.8); (5,.8) circle (.7mm); (0) at (0,-.6) [$1$]{}; (1) at (1,-.6) [$2$]{}; (3) at (4,-.6) [$p-1$]{}; (4) at (5,-.63) [$p$]{}; (5) at (6,-.6) [$p+1$]{}; (7) at (9,-.6) [$n$]{}; (empty) at (10,0) ;

Using a filtration by types of vertices of the opposition complex, we construct two exact chain complexes of $G$- and $G'$-modules. From these chain complexes, we obtain a spectral sequence involving relative homology of Levi subgroups with coefficient modules $M_p$ which are top-dimensional homology modules of opposition complexes of type $\{i_1,\ldots,i_{p-1}\}$, except for $M_1={\mathbb Z}$.

There is a spectral sequence with first page $$E^1_{p,q}=\begin{cases}
    H_q(G,G';{\mathbb Z}) & p=0 \\
    H_q(L_p,L'_p;M_p) & 1\leq p \leq n\\
    H_q(L_{n+1},G';M_{n+1}) & p=n+1
  \end{cases}$$ which converges to zero.

This can be used to prove homological stability for groups of type $A_{n+1}$ and $C_{n+1}$ in the following way: We want to prove that $H_q(G,G';{\mathbb Z})$ vanishes for all $q\leq k$ for a given $k$. Hence we must show that $H_q(L_p,L'_p;M_p)$ vanishes 