From Immersed Discs to Capped Gropes

'From Immersed Discs to Capped Gropes’ begins the proof of the disc embedding theorem in earnest. Starting with the immersed discs provided by the hypotheses of the disc embedding theorem, capped gropes with the same boundary and with suitable dual gropes are produced. This uses a sequence of the geometric moves introduced in the previous chapter. The two propositions in this chapter are technical, but vital. In subsequent chapters, the capped gropes will be upgraded to capped towers, and then to skyscrapers. The final step of the proof will consist of showing that skyscrapers are homeomorphic to the standard 2-handle, relative to the attaching region.