On homological stability for orthogonal and special orthogonal groups

Abstract

We shall prove that the map $H_i\left({\mathrm{SO}}_n\right(\mathbb{K}),\mathbb{Z})\to H_i\left({\mathrm{SO}}_{n+1}\right(\mathbb{K}),\mathbb{Z})$ is bijective for $2i<n$ and surjective for $2i\le n$. Here $\mathbb{K}$ is an arbitrary Pythagorean field and the special orthogonal group ${\mathrm{SO}}_n\left(\mathbb{K}\right)$ is the subgroup of $\mathbb{K}$-linear automorphisms over ${\mathbb{K}}^n$ with determinant one which preserve the Euclidean quadratic form $\mathbf{q}\left(x\right)=x_1^2+\cdots +x_n^2$. It is derived from the homological stability of the orthogonal groups ${\mathrm{O}}_n\left(\mathbb{K}\right)$ with twisted coefficients ${\mathbb{Z}}^t$.