Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Homological stability for classical groups


Authors: David Sprehn and Nathalie Wahl
Journal: Trans. Amer. Math. Soc. 373 (2020), 4807-4861
MSC (2010): Primary 20J05, 11E57
DOI: https://doi.org/10.1090/tran/8030
Published electronically: March 16, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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In addition, we include an exposition of Quillen's unpublished slope 1 stability argument for the general linear groups over fields other than $ \mathbb{F}_2$, and use it to recover also the improved range of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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 20J05, 11E57

Retrieve articles in all journals with MSC (2010): 20J05, 11E57


Additional Information

David Sprehn
Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
Email: david.sprehn@gmail.com

Nathalie Wahl
Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
Email: wahl@math.ku.dk

DOI: https://doi.org/10.1090/tran/8030
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).
Article copyright: © Copyright 2020 American Mathematical Society