Abstract
We prove a slope 1 stability range for the homology of the symplectic, orthogonal and unitary groups with respect to the hyperbolic form, over any fields other than $F_2$, improving the known range by a factor 2 in the case of finite fields. Our result more generally applies to the automorphism groups of vector spaces equipped with a possibly degenerate form (in the sense of Bak, Tits and Wall). For finite fields of odd characteristic, and more generally fields in which -1 is a sum of two squares, we deduce a stability range for the orthogonal groups with respect to the Euclidean form, and a corresponding result for the unitary groups. In addition, we include an exposition of Quillen's unpublished slope 1 stability argument for the general linear groups over fields other than $F_2$, and use it to recover also the improved range of Galatius-Kupers-Randal-Williams in the case of finite fields, at the characteristic.
Original language | English |
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Journal | Transactions of the American Mathematical Society |
Volume | 373 |
Issue number | 7 |
Pages (from-to) | 4807-4861 |
ISSN | 0002-9947 |
DOIs | |
Publication status | Published - 2020 |
Bibliographical note
v2: Revision. Now recovers the Galatius-Kupers-Randal-Williams improved stability range for general linear groups over finite fieldsKeywords
- math.AT
- math.KT