The Collar Adding Lemma

The collar adding lemma is a key ingredient in the proof of the disc embedding theorem. Specifically, it proves that a skyscraper with an added collar is homeomorphic to the standard 4-dimensional 2-handle. The proof is similar to the proof in a previous chapter that the Alexander gored ball with an added collar is homeomorphic to the standard 3-ball. Roughly speaking, a skyscraper is seen as the quotient space of the 4-ball corresponding to a certain decomposition. The added collar allows the decomposition to be modified so that the resulting decomposition shrinks; that is, the corresponding quotient space, which is identified with the skyscraper with an added collar, is homeomorphic to the original 4-ball.