On the groups of automorphisms of principal and fibre bundles

Abstract

If p: X → B is a principal G-bundle, an automorphism of p is an equivariant map of X to itself over B. The set G(p) of all such automorphisms inherits, in a natural way, a topological group structure. Similarly we can define, for a fibre bundle pF: XxGF → B, the group F(pF) of automorphisms of pF and, under suitable conditions, this is also a topological group. The purpose of this thesis is to obtain information on the homotopy properties of G(p) and F(PF). This is accomplished by using known relations between two bundles in order to determine corresponding relations between their groups of automorphisms. -- Having shown that F(PF) is, algebraically, a quotient of G(P) classified by the subgroup of G which acts trivially on F, we prove that such classification is often also topological. Moreover if h: G → K is a topological group morphism and ph is the bundle induced from p by h, there is a homomorphism Πhp: G(p) → G(ph), with image F(ph), which is a fibration if h is , or an n-equivalence if h has similar properties. This generates information on the fibre bundle problem and also on the effect of an enlargement of the structure group of p on G(p). Several computations are given, especially when the structure group is a classical group. -- The already known relation between G(p) and the space ΩMap(B,BG;k), where k is a classifying map for p, is then interpreted as a natural transformation connecting Πhp and the map induced by h, in the obvious way, between the corresponding loop spaces. We also outline a theory analogous to that of the main body of the thesis; in it a change of the base space replaces the change of the structure group or fibre. -- Finally, we give a non-standard construction of fibre bundles and associated principal bundles which leads to a simple proof of the equivalence between the categories of principal G-bundles over a space B and of fibre bundles over B with fibre an effective G-space F.