Chapter 21 - Classifying Spaces of Compact Lie Groups and Finite Loop Spaces

Abstract

This chapter deals with the explanation of classifying spaces of compact lie groups and finite loop spaces. The basic problem of the homotopy theory is to classify spaces and maps between spaces up to homotopy by means of algebraic invariants, such as homology or cohomology. Since their discovery, classifying spaces of compact Lie groups G, denoted by BG, are very important part in the homotopy theory. The chapter discusses various aspects of the theory. The chapter highlights the fact that for a simple connected compact Lie group G, two maps of BG are homotopic if they induce the same map in rational cohomology. The idea of developing the Lie group theory in terms of the homotopy theory goes back to Rector. In the analysis of maps between classifying spaces, decompositions into simpler pieces have proved to be quite a powerful tool. By simpler pieces, it means classifying spaces of subgroups. There are two different types of such decompositions. One uses centralizers of elementary abelian subgroups.