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:
https://doi.org/10.1090/S0002-9947-06-03917-1

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.