Homological stability for $\textrm {O}_ {n,n}$ over a local ring

Abstract:

Let $R$ be a local ring, ${V^{2n}}$ a free module over $R$ of rank $2n$ and $q$ a bilinear form on ${V^{2n}}$ which has in some basis the matrix $\left | {\begin {array}{*{20}{c}} 0 & 1 \\ 1 & 0 \\ \end {array} } \right |$. Let ${O_{n,n}}$ be the group of automorphisms of ${V^{2n}}$ which preserve $q$. We prove the following theorem: if $n$ is big enough with respect to $k$ then the inclusion homomorphism $i:{O_{n,n}} \to {O_{n + 1,n + 1}}$ induces an isomorphism ${i_{\ast }}:{H_k}({O_{n,n}}; Z) \to {H_k}({O_{n + 1,n + 1}};Z)$.