Homotopy uniqueness of classifying spaces of compact Lie groups

Anguru Kankananee Kirthi Premadasa, Purdue University

Abstract

This thesis deals with the following uniqueness conjecture. Let G be a compact Lie group and p be a prime. Conjecture. If X is a $F\sb{p}$-complete space such that $H\sp\*(X,F\sb{p})\cong H\sp\* (BG,F\sb{p})$ as algebras over $A\sb{p}$, the mod $-$p-Steenrod algebra, then, X is homotopy equivalent to $BG\sbsp{p}{\wedge}$. We outline a method to attack this problem in general and prove the conjecture for p = 2 for the cases BSO(5), BSO(7), and $BG\sb2$. We also introduce a technique which can be used to obtain the uniqueness of the classifying space of a compact Lie group, with knowledge of the uniqueness of its covering spaces. We use this to obtain the uniqueness of BSO(6), BO(6), BO(4) and others. Our method consists of decomposing X as the homotopy colimit of a functor and obtaining a map of diagrams between this functor and the corresponding funtor for $BG\sbsp{p}{\wedge}$. First, we identify the nodes of the two diagrams up to homotopy. Then we use results of Dwyer and Wilkerson to obtain a map of diagrams between the two funtors in the homotopy category. Finally, we remove the obstructions for the rigidifaction of this map, thus getting the desired map.

Degree

Ph.D.

Advisors

Wilkerson, Purdue University.

Subject Area

Mathematics

COinS