Pairings and monomorphisms of classifying spaces
type: Item
instance of: scholarly article
title: Pairings and monomorphisms of classifying spaces
zbMATH Open document ID: 1272.55011
author: Kenshi Ishiguro
author: Shotaro Kudo
author: Tomohiro Nakano
published in: Topology and its Applications
publication date: 2013-02-19T00:00:00Z
review text: Having fixed a monomorphism \(i: H \hookrightarrow G\) between compact Lie groups, the authors are interested in computing \((Bi)^\perp(BK, BG)\), the set of homotopy classes of maps \(\alpha: BK \rightarrow BG\) admitting a ``pairing'' \(BK \times BH \rightarrow BG\) restricting to \(Bi\) on \(BH\) and \(\alpha\) on \(BK\). When \(H\) is a semi-simple subgroup of a connected compact Lie group \(G\) of same rank, this set is trivial when \(K\) is connected. Explicit computations are performed for the inclusions \(i: SU(m) \subset SU(n)\) and \(j: Sp(m) \subset Sp(n)\). Here \[ (Bi)^\perp(BSU(k), BSU(n)) \cong [BSU(k), BSU(n-m)] \] and an analogous statement holds for \(j\). When \(n \geq 3\) and \(i: SO(n) \subset SU(n)\), then \[ (Bi)^\perp(BG, BSU(n)) = 0 \] for any connected compact Lie group \(G\).
reviewed by: Jérôme Scherer
Mathematics Subject Classification ID: 55R37
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55P60
zbMATH DE Number: 6137373
zbMATH Keywords: classifying space
zbMATH Keywords: Lie group
zbMATH Keywords: \(p\)-compact group
zbMATH Keywords: \(p\)-completion
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: admissible map
MaRDI profile type: Publication
full work available at URL: j.topol.2012.10.018
OpenAlex ID: W2061914345
DOI: 10.1016/J.TOPOL.2012.10.018
description: scientific article; zbMATH DE number 6137373