On extensions of principal bundles
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 principle bundles, gives a new concept of equivariance, unrelated to the standard concept of equivariant bundle as treated by, for example, \textit{R. K. Lashof, J. P. May} and \textit{G. B. Segal} [Contemp. Math. 19, 167-176 (1983; Zbl 0526.55020)]. In Section 3 we apply this concept of equivariance to connections in Q(P,N,\(\pi)\) and introduce an equivariant Chern-Weil morphism from the H-invariant symmetric forms on the Lie algebra of N, to the G-invariant de Rham cohomology of P. We show that when all three groups are compact and connected, the Chern-Weil classes of Q(B,H) can be expressed in terms of those of P(B,G) together with the equivariant classes of Q(P,N). This compact-connected case is quite special, and simple, and we expect the method to be much more generally applicable.
scientific article; zbMATH DE number 4019996