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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Whitney towers and gropes in 4-manifolds


Author: Rob Schneiderman
Journal: Trans. Amer. Math. Soc. 358 (2006), 4251-4278
MSC (2000): Primary 57M99; Secondary 57M25
Published electronically: June 21, 2005
MathSciNet review: 2231378
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Abstract: Many open problems and important theorems in low-dimensional topology have been formulated as statements about certain 2-complexes called gropes. This paper describes a precise correspondence between embedded gropes in 4-manifolds and the failure of the Whitney move in terms of iterated `towers' of Whitney disks. The `flexibility' of these Whitney towers is used to demonstrate some geometric consequences for knot and link concordance connected to $n$-solvability, $k$-cobordism and grope concordance. The key observation is that the essential structure of gropes and Whitney towers can be described by embedded unitrivalent trees which can be controlled during surgeries and Whitney moves. It is shown that a Whitney move in a Whitney tower induces an IHX (Jacobi) relation on the embedded trees.


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Additional Information

Rob Schneiderman
Affiliation: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012-1185
Email: schneiderman@courant.nyu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03768-2
PII: S 0002-9947(05)03768-2
Keywords: Whitney tower, grope, 4--manifold, Whitney move, $n$-solvability, $k$-cobordism, grope concordance, IHX relation
Received by editor(s): March 7, 2004
Received by editor(s) in revised form: May 25, 2004
Published electronically: June 21, 2005
Additional Notes: The author is an NSF VIGRE postdoctoral fellow at the Courant Institute of Mathematical Sciences.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.