Hyperbolic posets and homology stability for \(O_{n,n}\)
type: Item
instance of: scholarly article
title: Hyperbolic posets and homology stability for \(O_{n,n}\)
zbMATH Open document ID: 0604.20045
DOI: 10.1016/0022-4049(86)90001-0
author: Stanislaw Betley
published in: Journal of Pure and Applied Algebra
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 55U10
Mathematics Subject Classification ID: 18G40
Mathematics Subject Classification ID: 57T10
Mathematics Subject Classification ID: 51F25
Mathematics Subject Classification ID: 51D25
zbMATH DE Number: 3975218
zbMATH Keywords: orthogonal group of quadratic form
zbMATH Keywords: hyperbolic space
zbMATH Keywords: inclusion map
zbMATH Keywords: homology
zbMATH Keywords: poset of hyperbolic subspaces
MaRDI profile type: Publication
full work available at URL: 0022-4049(86)90001-0
OpenAlex ID: W2044814083
cites work: Q4043127
cites work: Algebraic L -Theory, I: Foundations
cites work: Spherical posets and homology stability for \(O_ n,\)//n
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description: scientific article; zbMATH DE number 3975218