Maps between classifying spaces of Kac--Moody groups
type: Item
instance of: scholarly article
title: Maps between classifying spaces of Kac--Moody groups
zbMATH Open document ID: 1029.55013
DOI: 10.1016/S0001-8708(02)00065-8
author: Jaume Aguadé
author: Albert Ruiz
published in: Advances in Mathematics
publication date: 2003-08-25T00:00:00Z
review text: Kac-Moody groups are obtained from Kac--Moody algebras in a way that generalises the usual construction of Lie groups from finite dimensional Lie algebras. In this paper the authors consider the unitary form of a non-affine rank-two Kac--Moody group \(K\). For these groups, the classifying space \(BK\) is a push-out of classifying spaces of compact Lie groups. The authors use various homotopy theoretic methods to study the homotopy classes of self-maps of \(BK\). The integral cohomology of \(BK\) is used to define the degree of a self-map of \(BK\). The authors construct \textit{generic Adams maps} and show that every self-map of \(BK\) is homotopy equivalent to a generic Adams map. They show that self-maps are determined up to homotopy by their restriction to the classifying space of a maximal torus and the authors give a description of the monoid \([BK,BK]\). While some aspects of the homotopy theory of compact Lie groups generalise to the Kac-Moody groups studied here, there are notable exceptions. In Section~6, it is shown that there are non-isomorphic Kac-Moody groups with homotopically equivalent classifying spaces. This is in contrast to the fact that two connected compact Lie groups are isomorphic if and only if their classifying spaces are homotopy equivalent [\textit{D. Notbohm} J. Lond. Math. Soc., II. Ser. 52, 185-198 (1995; Zbl 0836.55008)]. The authors also show that, for a torus \(T\), the natural map from \(Hom(T,K)\) to \([BT,BK]\) is not necessarily surjective (Remark 3.4). For a compact Lie group, the map from \(Hom(T,K)\) to \([BT,BK]\) is surjective (see [\textit{D. Notbohm}, Maps between classifying spaces, Univ. Goettingen, Diss. (1988; Zbl 0675.55009)].
reviewed by: R. J. Shank
Mathematics Subject Classification ID: 55R37
Mathematics Subject Classification ID: 81R10
zbMATH DE Number: 1970566
zbMATH Keywords: classifying space
zbMATH Keywords: Kac-Moody group
zbMATH Keywords: self-maps
MaRDI profile type: Publication
cites work: Constructing modular classifying spaces
cites work: Cohomology of classifying spaces of central quotients of rank two Kac-Moody groups
cites work: The cohomology of quotients of classical groups
cites work: Homotopy limits, completions and localizations
cites work: Classifying spaces of Kač-Moody groups
cites work: Q4177654
cites work: Q3790321
cites work: Unstable Adams operations on classifying spaces
cites work: Q4734492
cites work: Homotopy classification of self-maps of \(BG\) via \(G\)-actions. II
cites work: Q4851842
cites work: Q4280931
cites work: Q3859954
cites work: Q3797354
cites work: On automorphisms of Kac-Moody algebras and groups
cites work: On the topology of Kac-Moody groups
cites work: Q3830229
cites work: On the ‘Classifying Space’ Functor for Compact Lie Groups
cites work: Q4896557
cites work: Infinite flag varieties and conjugacy theorems
cites work: Q4084951
full work available at URL: s0001-8708(02)00065-8
OpenAlex ID: W1988209552
Recommended article: On the topology of Kac-Moody groups
Recommended article: On some applications of unstable Adams operations to the topology of Kac-Moody groups
Recommended article: Homology of the Kač-Moody groups. I
Recommended article: The rational cohomology Hopf algebra of a generic Kac-Moody group
Recommended article: Classifying spaces of Kač-Moody groups
Recommended article: Cohomology of classifying spaces of central quotients of rank two Kac-Moody groups
Recommended article: Q4539658
Recommended article: Computation of the cohomology rings of Kac-Moody groups, their flag manifolds and classifying spaces
Recommended article: Q5209130
Recommended article: Discrete approximations for complex Kac-Moody groups
description: scientific article; zbMATH DE number 1970566