String topology of finite groups of Lie type

Publikation: Working paperPreprintForskning

Standard

String topology of finite groups of Lie type. / Grodal, Jesper; Lahtinen, Anssi.

arxiv.org, 2020.

Publikation: Working paperPreprintForskning

Harvard

Grodal, J & Lahtinen, A 2020 'String topology of finite groups of Lie type' arxiv.org. <https://arxiv.org/pdf/2003.07852v1.pdf>

APA

Grodal, J., & Lahtinen, A. (2020). String topology of finite groups of Lie type. arxiv.org. https://arxiv.org/pdf/2003.07852v1.pdf

Vancouver

Grodal J, Lahtinen A. String topology of finite groups of Lie type. arxiv.org. 2020.

Author

Grodal, Jesper ; Lahtinen, Anssi. / String topology of finite groups of Lie type. arxiv.org, 2020.

Bibtex

@techreport{e0d86c42bb88418c834364774cab787b,
title = "String topology of finite groups of Lie type",
abstract = " We show that the mod $\ell$ cohomology of any finite group of Lie type in characteristic $p$ different from $\ell$ admits the structure of a module over the mod $\ell$ cohomology of the free loop space of the classifying space $BG$ of the corresponding compact Lie group $G$, via ring and module structures constructed from string topology, a la Chas-Sullivan. If a certain fundamental class in the homology of the finite group of Lie type is non-trivial, then this module structure becomes free of rank one, and provides a structured isomorphism between the two cohomology rings equipped with the cup product, up to a filtration. We verify the nontriviality of the fundamental class in a range of cases, including all simply connected untwisted classical groups over the field of $q$ elements, with $q$ congruent to 1 mod $\ell$. We also show how to deal with twistings and get rid of the congruence condition by replacing $BG$ by a certain $\ell$-compact fixed point group depending on the order of $q$ mod $\ell$, without changing the finite group. With this modification, we know of no examples where the fundamental class is trivial, raising the possibility of a general structural answer to an open question of Tezuka, who speculated about the existence of an isomorphism between the two cohomology rings. ",
keywords = "math.AT, math.GR, math.RT, 20J06 (Primary) 20D06, 55R35, 55P50 (Secondary)",
author = "Jesper Grodal and Anssi Lahtinen",
note = "58 pages",
year = "2020",
language = "English",
publisher = "arxiv.org",
type = "WorkingPaper",
institution = "arxiv.org",

}

RIS

TY - UNPB

T1 - String topology of finite groups of Lie type

AU - Grodal, Jesper

AU - Lahtinen, Anssi

N1 - 58 pages

PY - 2020

Y1 - 2020

N2 - We show that the mod $\ell$ cohomology of any finite group of Lie type in characteristic $p$ different from $\ell$ admits the structure of a module over the mod $\ell$ cohomology of the free loop space of the classifying space $BG$ of the corresponding compact Lie group $G$, via ring and module structures constructed from string topology, a la Chas-Sullivan. If a certain fundamental class in the homology of the finite group of Lie type is non-trivial, then this module structure becomes free of rank one, and provides a structured isomorphism between the two cohomology rings equipped with the cup product, up to a filtration. We verify the nontriviality of the fundamental class in a range of cases, including all simply connected untwisted classical groups over the field of $q$ elements, with $q$ congruent to 1 mod $\ell$. We also show how to deal with twistings and get rid of the congruence condition by replacing $BG$ by a certain $\ell$-compact fixed point group depending on the order of $q$ mod $\ell$, without changing the finite group. With this modification, we know of no examples where the fundamental class is trivial, raising the possibility of a general structural answer to an open question of Tezuka, who speculated about the existence of an isomorphism between the two cohomology rings.

AB - We show that the mod $\ell$ cohomology of any finite group of Lie type in characteristic $p$ different from $\ell$ admits the structure of a module over the mod $\ell$ cohomology of the free loop space of the classifying space $BG$ of the corresponding compact Lie group $G$, via ring and module structures constructed from string topology, a la Chas-Sullivan. If a certain fundamental class in the homology of the finite group of Lie type is non-trivial, then this module structure becomes free of rank one, and provides a structured isomorphism between the two cohomology rings equipped with the cup product, up to a filtration. We verify the nontriviality of the fundamental class in a range of cases, including all simply connected untwisted classical groups over the field of $q$ elements, with $q$ congruent to 1 mod $\ell$. We also show how to deal with twistings and get rid of the congruence condition by replacing $BG$ by a certain $\ell$-compact fixed point group depending on the order of $q$ mod $\ell$, without changing the finite group. With this modification, we know of no examples where the fundamental class is trivial, raising the possibility of a general structural answer to an open question of Tezuka, who speculated about the existence of an isomorphism between the two cohomology rings.

KW - math.AT

KW - math.GR

KW - math.RT

KW - 20J06 (Primary) 20D06, 55R35, 55P50 (Secondary)

M3 - Preprint

BT - String topology of finite groups of Lie type

PB - arxiv.org

ER -

ID: 311260766