A looping-delooping adjunction for topological spaces

Abstract

Every principal G-bundle is classified up to equivalence by a homotopy class of maps into the classifying space of G. On the other hand, for every nice topological space Milnor constructed a strict model of loop space, that is a group. Moreover the morphisms of topological groups defined on the loop space of X generate all the bundles over X up to equivalence. In this paper, we show that the relationship between Milnor's loop space and the classifying space functor is, in a precise sense, an adjoint pair between based spaces and topological groups in a homotopical context. This proof leads to a classification of principal bundles with a fixed structure group. Such a resul clarifies the deep relation that exists between the theory of bundles, the classifying space construction and the loop space construction, which are very important in topological K-theory, group cohomology and homotopy theory.