Homological stability for $\textrm {O}_ {n,n}$ over a local ring
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- by Stanisลaw Betley PDF
- Trans. Amer. Math. Soc. 303 (1987), 413-429 Request permission
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 ${i_{\ast }}:{H_k}({O_{n,n}}; Z) \to {H_k}({O_{n + 1,n + 1}};Z)$.References
- Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
- Stanisลaw Betley, Hyperbolic posets and homology stability for $O_{n,n}$, J. Pure Appl. Algebra 43 (1986), no.ย 1, 1โ9. MR 862869, DOI 10.1016/0022-4049(86)90001-0
- Armand Borel, Stable real cohomology of arithmetic groups, Ann. Sci. รcole Norm. Sup. (4) 7 (1974), 235โ272 (1975). MR 387496, DOI 10.24033/asens.1269
- Ruth M. Charney, Homology stability for $\textrm {GL}_{n}$ of a Dedekind domain, Invent. Math. 56 (1980), no.ย 1, 1โ17. MR 557579, DOI 10.1007/BF01403153
- W. G. Dwyer, Twisted homological stability for general linear groups, Ann. of Math. (2) 111 (1980), no.ย 2, 239โ251. MR 569072, DOI 10.2307/1971200
- Wilberd van der Kallen, Homology stability for linear groups, Invent. Math. 60 (1980), no.ย 3, 269โ295. MR 586429, DOI 10.1007/BF01390018 H. Maazen, Homology stability for the general linear group, Thesis, Univ. of Utrecht, 1979. D. Quillen, MIT Lectures, 1974-75.
- A. A. Ranicki, Algebraic $L$-theory. I. Foundations, Proc. London Math. Soc. (3) 27 (1973), 101โ125. MR 414661, DOI 10.1112/plms/s3-27.1.101
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112 A. A. Suslin, Stability in algebraic $K$-theory, Lecture Notes in Math., vol. 966, Springer-Verlag, Berlin and New York, 1982.
- Karen Vogtmann, Homology stability for $\textrm {O}_{n,n}$, Comm. Algebra 7 (1979), no.ย 1, 9โ38. MR 514863, DOI 10.1080/00927877908822331
- Karen Vogtmann, Spherical posets and homology stability for $\textrm {O}_{n,n}$, Topology 20 (1981), no.ย 2, 119โ132. MR 605652, DOI 10.1016/0040-9383(81)90032-X
- K. Vogtmann, A Stiefel complex for the orthogonal group of a field, Comment. Math. Helv. 57 (1982), no.ย 1, 11โ21. MR 672842, DOI 10.1007/BF02565843
- J. B. Wagoner, Stability for homology of the general linear group of a local ring, Topology 15 (1976), no.ย 4, 417โ423. MR 417263, DOI 10.1016/0040-9383(76)90035-5
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 303 (1987), 413-429
- MSC: Primary 20G10; Secondary 11E72, 18G99, 19D55
- DOI: https://doi.org/10.1090/S0002-9947-1987-0896030-4
- MathSciNet review: 896030