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The space of commuting elements in a Lie group and maps between classifying spaces

Part of: Fiber spaces and bundles Homology and homotopy of topological groups and related structures

Published online by Cambridge University Press:  27 October 2023

Daisuke Kishimoto ,
Masahiro Takeda  and
Mitsunobu Tsutaya
Daisuke Kishimoto
Affiliation:
Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan (kishimoto@math.kyushu-u.ac.jp)
Masahiro Takeda
Affiliation:
Institute for Liberal Arts and Sciences, Kyoto University, Kyoto 606-8316, Japan (takeda.masahiro.87u@kyoto-u.ac.jp)
Mitsunobu Tsutaya
Affiliation:
Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan (tsutaya@math.kyushu-u.ac.jp)
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Abstract

Let $\pi$ be a discrete group, and let $G$ be a compact-connected Lie group. Then, there is a map $\Theta \colon \mathrm {Hom}(\pi,G)_0\to \mathrm {map}_*(B\pi,BG)_0$ between the null components of the spaces of homomorphisms and based maps, which sends a homomorphism to the induced map between classifying spaces. Atiyah and Bott studied this map for $\pi$ a surface group, and showed that it is surjective in rational cohomology. In this paper, we prove that the map $\Theta$ is surjective in rational cohomology for $\pi =\mathbb {Z}^m$ and the classical group $G$ except for $SO(2n)$, and that it is not surjective for $\pi =\mathbb {Z}^m$ with $m\ge 3$ and $G=SO(2n)$ with $n\ge 4$. As an application, we consider the surjectivity of the map $\Theta$ in rational cohomology for $\pi$ a finitely generated nilpotent group. We also consider the dimension of the cokernel of the map $\Theta$ in rational homotopy groups for $\pi =\mathbb {Z}^m$ and the classical groups $G$ except for $SO(2n)$.

Keywords

space of commuting elementsLie groupclassifying spaceflat connectionrational cohomology

MSC classification

Primary: 55R37: Maps between classifying spaces
Secondary: 57T10: Homology and cohomology of Lie groups
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 21
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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