Notes on principal bundles and classifying spaces

Consider a real n-plane bundle ξ with Euclidean metric. Associated to ξ are a number of auxiliary bundles: disc bundle, sphere bundle, projective bundle, k-frame bundle, etc. Here “bundle” simply means a local product with the indicated fibre. In each case one can show, by easy but repetitive arguments, that the projection map in question is indeed a local product; furthermore, the transition functions are always linear in the sense that they are induced in an obvious way from the linear transition functions of ξ. It turns out that all of this data can be subsumed in a single object: the “principal O(n)-bundle” Pξ, which is just the bundle of orthonormal n-frames. The fact that the transition functions of the various associated bundles are linear can then be formalized in the notion “fibre bundle with structure group O(n)”. If we do not want to consider a Euclidean metric, there is an analogous notion of principal GLnR-bundle; this is the bundle of linearly independent n-frames. More generally, if G is any topological group, a principal G-bundle is a locally trivial free G-space with orbit space B (see below for the precise definition). For example, if G is discrete then a principal G-bundle with connected total space is the same thing as a regular covering map with G as group of deck transformations. Under mild hypotheses there exists a classifying space BG, such that isomorphism classes of principal G-bundles over X are in natural bijective correspondence with [X,BG]. The correspondence is given by pulling back a universal principal G-bundle over BG. When G is discrete, BG is an Eilenberg-Maclane space of type (G, 1). When G is either GLnR or O(n), BG is homotopy equivalent to the infinite Grassmanian GnR. The homotopy classification theorem for vector bundles then emerges as a special case of the homotopy classification theorem for principal bundles. As these examples begin to suggest, the concept principal bundle acts as a powerful unifying force in algebraic topology. Classifying spaces also play a central role; indeed, much of the research in homotopy theory over the last fifty years involves analyzing the homotopytype of BG for interesting groups G. There are also many applications in differential geometry, involving connections, curvature, etc. In these notes we will study principal bundles and classifying spaces from the homotopy-theoretic point of view.