On the equivariant Whitehead towers of BO and BU.
Summary
The main focus of this research project will be constructing and studying the structure of the Whitehead towers for the classifying spaces of the stable orthogonal and unitary groups denoted by BO and BU respectively. Given an arbitrary topological space X we look at the obstructions for lifting a map X → BU⟨n⟩ to a map X → BU⟨n+1⟩ along the tower, where BU⟨n⟩ denotes the nth stage of the Whitehead tower, and relate these obstructions to characteristic classes. Analogously we study the lifts of maps X → BO⟨n⟩. Moreover we will study the dual construction, namely the Postnikov tower. First we work on proving the existence of a bijection between symmetric bilinear forms over the integers and homotopy classes of maps CP^∞ → P^4 BO, where P^n BO denotes the nth stage of the Postnikov tower of BO. Finally we study the more general case of homotopy classes of maps (CP^∞)^d → P^n BU and (CP^∞)^d → P^n BO.