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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Homological stability for $ {\rm O}\sb {n,n}$ over a local ring


Author: Stanisław Betley
Journal: Trans. Amer. Math. Soc. 303 (1987), 413-429
MSC: Primary 20G10; Secondary 11E72, 18G99, 19D55
MathSciNet review: 896030
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Abstract | References | Similar Articles | Additional Information

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\vert {\begin{array}{*{20}{c}} 0 & 1 \\ 1 & 0 \\ \end{array} } \right\vert\,$. 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)$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0896030-4
PII: S 0002-9947(1987)0896030-4
Keywords: Spherical simplicial complex, homology of a group
Article copyright: © Copyright 1987 American Mathematical Society