The Whitehead Decomposition

Abstract

‘The Whitehead Decomposition’ introduces this historically significant decomposition. Not only is the quotient of the 3-sphere by the Whitehead decomposition not homeomorphic to the 3-sphere, it is not even a manifold. In order to detect this curious fact, the notion of a noncompact space being simply connected at infinity is introduced. The chapter also describes the Whitehead manifold, which is a contractible 3-manifold not homeomorphic to Euclidean space. While the Whitehead decomposition does not shrink, its product with the real line does, as is proved in this chapter; in other words, the quotient of the 3-sphere by the Whitehead decomposition is a manifold factor. The proof of the disc embedding theorem utilizes Bing–Whitehead decompositions, which may be understood to be a mix between the Whitehead decomposition and the Bing decomposition from a previous chapter. In a subsequent chapter, precisely when Bing–Whitehead decompositions shrink is explained.