Homological stability for classical groups
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Homological stability for classical groups. / Sprehn, David; Wahl, Nathalie.
In: Transactions of the American Mathematical Society, Vol. 373, No. 7, 2020, p. 4807-4861.Research output: Contribution to journal › Journal article › Research › peer-review
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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 In addition, we include an exposition of Quillen's unpublished slope 1 stability argument for the general linear groups over fields other than $F_2$, and use it to recover also the improved range of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 In addition, we include an exposition of Quillen's unpublished slope 1 stability argument for the general linear groups over fields other than $F_2$, and use it to recover also the improved range of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