## Homological stability for classical groups

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**Homological stability for classical groups.** / Sprehn, David; Wahl, Nathalie.

Research output: Contribution to journal › Journal article › Research › peer-review

#### Harvard

*Transactions of the American Mathematical Society*, vol. 373, no. 7, pp. 4807-4861. https://doi.org/10.1090/tran/8030

#### APA

*Transactions of the American Mathematical Society*,

*373*(7), 4807-4861. https://doi.org/10.1090/tran/8030

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#### Bibtex

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#### RIS

TY - JOUR

T1 - Homological stability for classical groups

AU - Sprehn, David

AU - Wahl, Nathalie

N1 - v2: Revision. Now recovers the Galatius-Kupers-Randal-Williams improved stability range for general linear groups over finite fields

PY - 2020

Y1 - 2020

N2 - 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 In addition, we include an exposition of Quillen's unpublished slope 1 stability argument for the general linear groups over fields other than $F_2$, and use it to recover also the improved range of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.

AB - 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.

KW - math.AT

KW - math.KT

U2 - 10.1090/tran/8030

DO - 10.1090/tran/8030

M3 - Journal article

VL - 373

SP - 4807

EP - 4861

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 7

ER -

ID: 248189976