The Homotopy Type of G/TOP, G/PL and G/O

In this chapter we review how to determine the homotopy types of the spaces G/PL and G/TOP and discuss some related results. We have already seen the statements in Section 11.9, here we deal with the proofs. Both the statements and the proofs are due to Sullivan in the PL case. In the topological case they are also due to Sullivan, but they are based on the foundational work of Kirby and Siebenmann [219]. The PL case is an input in the work of Kirby and Siebenmann, hence the topological case is a corollary of the PL case in this sense. However, we will concentrate on the topological case since there the statement is simpler, and we will rely the work of Kirby and Siebenmann without further explanations.