On extensions of principal bundles
type: Item
instance of: scholarly article
title: On extensions of principal bundles
zbMATH Open document ID: 0627.55010
DOI: 10.1007/BF00133036
published in: Annals of Global Analysis and Geometry
publication date: 1988-01-01T00:00:00Z
review text: By an extension of principal bundles we mean a system \[ N\quad \rightarrowtail \quad Q(B,H,q)\twoheadrightarrow ^{\pi}P(B,G,p) \] in which P(B,G,p) and Q(B,H,q) are principal bundles and \(\pi\) : \(Q\to P\), \(\pi\) : \(H\to G\) is a morphism of principal bundles over B and a surjective submersion, and in which N is a (possibly nonabelian) Lie group and \(N\to H\) is an injection of Lie groups making \(N\rightarrowtail H\twoheadrightarrow G\) exact. When P(B,G) and H are fixed and Q is sought, this is the problem of lifting a structure group [for which see, for example, \textit{A. Haefliger}, C. R. Acad. Sci., Paris 243, 558-560 (1956; Zbl 0070.400)]; we are concerned with the situation in which only P(B,G) and N are given and both Q and H are sought. The paper is founded on the observation that the group G, although it does not usually act on H or Q, does in fact act naturally on the Lie groupoid \(\gamma =(Q\times Q)/N\) associated to the ``transverse bundle'' Q(P,N,\(\pi)\). Further, given P(B,G), and a Lie groupoid \(\gamma\) on P on which G acts by Lie groupoid automorphisms over the principal action \(P\times G\to P\), the quotient manifold \(\gamma\) /G exists and Q(B,H) can be recovered from it. These results are from the author's paper [Cah. Topologie Géom. Différ. Categoriques 28, 29-52 (1987)] and are recalled in Section 1 with the necessary minimum of background on Lie groupoids. This equivalence between extensions of P(B,G) and Lie groupoids \(\gamma\) on P equipped with a G-action of this type allows us to prove in Section 2 that if N is discrete abelian and P is connected then the group of those equivalence classes of extensions of P(B,G) by N which induce a chosen action of G on N, is isomorphic to \(Hom(\pi _{ab},N)^ G\), where \(\pi _{ab}\) is abelianized \(\pi _ 1(P)\) and G acts on \(\pi _{ab}\) and N in the evident ways. Most bundle theory can be formulated in terms of Lie groupoids, and the action of G on \(\gamma\), in a general extension of…
Mathematics Subject Classification ID: 55R10
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 20L05
Mathematics Subject Classification ID: 55R91
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 53C05
zbMATH DE Number: 4019996
zbMATH Keywords: extension of principal bundles
zbMATH Keywords: morphisms of principal bundles
zbMATH Keywords: lifting a structure group
zbMATH Keywords: Lie groupoid
zbMATH Keywords: transverse bundle
author: Kirill Charles Howard Mackenzie
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4019996