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The Disc Embedding Theorem$
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Stefan Behrens, Boldizsar Kalmar, Min Hoon Kim, Mark Powell, and Arunima Ray

Print publication date: 2021

Print ISBN-13: 9780198841319

Published to Oxford Scholarship Online: September 2021

DOI: 10.1093/oso/9780198841319.001.0001

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PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 04 December 2021

The Ball to Ball Theorem

The Ball to Ball Theorem

(p.131) 10 The Ball to Ball Theorem
The Disc Embedding Theorem

Stefan Behrens

Boldizsár Kalmár

Daniele Zuddas

Oxford University Press

The ball to ball theorem is presented, which states that a map from the 4-ball to itself, restricting to a homeomorphism on the 3-sphere, whose inverse sets are null and have nowhere dense image, is approximable by homeomorphisms relative to the boundary. The approximating homeomorphisms are produced abstractly, as in the previous chapter, with no need to investigate the decomposition elements further. In the proof of the disc embedding theorem, a decomposition of the 4-ball will be constructed, called the gaps+ decomposition. The ball to ball theorem will be used to prove that this decomposition shrinks; this is called the β-shrink.

Keywords:   ball to ball theorem, null decomposition, β-shrink, disc embedding theorem, gaps+ decomposition

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