F-fibrations and groups of gauge transformations

Abstract

A study of the relationships between various notions of "Universal fibration" which arise in the literature, has been done by P. Booth, P. Heath and R. Piccinini, within the context of an admissible category of fibrations. This general category, of which the usual categories of fibrations that arise in practise are particular examples, is defined within the general framework introduced by J. P. May for the notion of F- fibration, by specifying certain axioms. Using a generalized version of the exponential law we show that the category of F- fibrations is directly related to this notion of an admissible category of fibrations. The result is that the axioms defining admissibility can be simplified. -- The appropriate notion of equivalence in an admissible category A is an extension of the notion of fibre homotopy equivalence, called F - homotopy equivalence. If p: E → B is an A- fibration (object of A), we denote by F(p), the space of all F- homotopy equivalences p → p over B and by F¹(p), the space of all F- homotopy equivalences p → p over B which extend 1: F → F on a distinguished fibre F. We show that if the category A admits an "Aspherical Universal" A- fibration P∞: E∞ → B∞ (this is the situation in the usual categories of fibrations that arise in practise) and k: B → B∞ is the classifying map for p, then F(p) (resp. F¹(p)) has the same weak homotopy type as ΩL(B,B∞;k)(resp. ΩL*(B, B∞;k)). Here, L(B,B∞;k) denotes the path component of the function space L(B,B∞) which contains k and L*(B,B∞;k) is the based version. In particular, we show that if B∞ is an H-group then F(p) (resp. F¹(p)) has the same weak homotopy type as L(B,ΩB∞ (resp. L*(B,ΩB∞)); if B is an H-cogroup, then F¹(p) has the same weak homotopy type as L*(B,ΩB∞). With a connectivity condition on B∞ it is also possible to obtain some computations of the homotopy groups of F(p) and F¹(p) within the stable range. In the case where the admissible category A is the category of principal G-bundles over smooth manifolds, with G a compact Lie, group the spaces F(p) and F¹(p) are the groups of gauge transformations that arise in theoretical Physics. The results obtained in the general situation are valid here up to homotopy equivalence.