A looping-delooping adjunction for topological spaces

TL;DR

This paper establishes a homotopical adjunction between Milnor's loop space and the classifying space functor, providing a new perspective on classifying principal bundles and their relation to topological groups.

Contribution

It demonstrates that Milnor's loop space and the classifying space form an adjoint pair, leading to a classification of principal bundles with fixed structure groups.

Findings

The relationship between loop spaces and classifying spaces is an adjoint pair.

This adjunction clarifies the connection between bundle theory and homotopy theory.

Provides a new framework for classifying principal G-bundles.

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.