A looping–delooping adjunction for topological spaces

Every principal $G$-bundle over $X$ is classified up to equivalence by a homotopy class $X \to BG$, where $BG$ is the classifying space of $G$. On the other hand, for every nice topological space $X$ Milnor constructed a strict model of its loop space $\tilde{\Omega}X$, that is a group. Moreover, the morphisms of topological groups $\tilde{\Omega}X \to G$ generate all the $G$-bundles over $X$ up to equivalence.

In this paper, we show that the relation 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 over a fixed space, that is dual to the classification of bundles with a fixed group. Such a result clarifies the deep relation that exists between the theory of bundles, the classifying space construction and the loop space, which are very important in topological $K$-theory, group cohomology, and homotopy theory.

Keywords

principal bundle, loop space, classifying space

2010 Mathematics Subject Classification

55P35, 55R15, 55R35

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Published 6 June 2017