Homological stability for classical groups
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- by David Sprehn and Nathalie Wahl PDF
- Trans. Amer. Math. Soc. 373 (2020), 4807-4861 Request permission
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 $\mathbb {F}_2$, 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In addition, we include an exposition of Quillen’s unpublished slope 1 stability argument for the general linear groups over fields other than $\mathbb {F}_2$, and use it to recover also the improved range of Galatius–Kupers–Randal-Williams in the case of finite fields, at the characteristic.
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Additional Information
- David Sprehn
- Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
- MR Author ID: 1094380
- Email: david.sprehn@gmail.com
- Nathalie Wahl
- Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
- MR Author ID: 677027
- ORCID: 0000-0003-2063-9234
- Email: wahl@math.ku.dk
- Received by editor(s): February 4, 2019
- Received by editor(s) in revised form: August 7, 2019, and October 15, 2019
- Published electronically: March 16, 2020
- Additional Notes: The authors were supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).
The second author was also supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 772960), the Heilbronn Institute for Mathematical Research, and the Isaac Newton Institute for Mathematical Sciences, Cambridge (EPSRC grant numbers EP/K032208/1 and EP/R014604/1). - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 4807-4861
- MSC (2010): Primary 20J05, 11E57
- DOI: https://doi.org/10.1090/tran/8030
- MathSciNet review: 4127863