Hyperbolic posets and homology stability for \(O_{n,n}\)
Let O(n,n) denote the orthogonal group of the following quadratic form, where 0,1 are the n by n zero and identity matrices respectively, on a 2n-dimensional vector space over a field F (such an endowed V is called a hyperbolic space): \(\left( \begin{matrix} 0\quad 1\\ 1\quad 0\end{matrix} \right).\) \textit{K. Vogtmann} proved [in Topology 20, 119-132 (1981; Zbl 0455.20031)] that the inclusion map \(O(n,n)\to O(n+1,n+1)\) induces an isomorphism on the homology in dimension k if \(n\geq 3k\). The current paper reproves this result, although with 3k replaced by \(3k+3\), using a poset of hyperbolic subspaces of V instead of Vogtmann's poset of isotropic subspaces.
scientific article; zbMATH DE number 3975218
scientific article