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

 

Dual decompositions of 4-manifolds III: s-cobordisms


Author: Frank Quinn
Journal: Trans. Amer. Math. Soc. 359 (2007), 1433-1443
MSC (2000): Primary 57N13, 57N70, 57R80
Published electronically: August 16, 2006
MathSciNet review: 2272132
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Abstract: The main result is that an s-cobordism (topological or smooth) of 4-manifolds has a product structure outside a ``core'' sub-s-cobordism. These cores are arranged to have quite a bit of structure, for example they are smooth and abstractly (forgetting boundary structure) diffeomorphic to a standard neighborhood of a 1-complex. The decomposition is highly nonunique so cannot be used to define an invariant, but it shows that the topological s-cobordism question reduces to the core case. The simply-connected version of the decomposition (with 1-complex a point) is due to Curtis, Freedman, Hsiang and Stong. Controlled surgery is used to reduce topological triviality of core s-cobordisms to a question about controlled homotopy equivalence of 4-manifolds. There are speculations about further reductions. The decompositions on the ends of the s-cobordism are ``dual decompositions'' with homotopically-controlled handle structures, and the main result is an application of earlier papers in the series.


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Additional Information

Frank Quinn
Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061-0123
Email: quinn@math.vt.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03917-1
PII: S 0002-9947(06)03917-1
Received by editor(s): September 24, 2004
Received by editor(s) in revised form: November 30, 2004
Published electronically: August 16, 2006
Additional Notes: This work was partially supported by the National Science Foundation
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.



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