Dual decompositions of 4-manifolds

Author:
Frank Quinn

Journal:
Trans. Amer. Math. Soc. **354** (2002), 1373-1392

MSC (2000):
Primary 57R65, 57M20

Published electronically:
November 8, 2001

MathSciNet review:
1873010

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Abstract: This paper concerns decompositions of smooth 4-manifolds as the union of two handlebodies, each with handles of index . In dimensions results of Smale (trivial ) and Wall (general ) describe analogous decompositions up to diffeomorphism in terms of homotopy type of skeleta or chain complexes. In dimension 4 we show the same data determines decompositions up to 2-deformation of their spines. In higher dimensions spine 2-deformation implies diffeomorphism, but in dimension 4 the fundamental group of the boundary is not determined. Sample results: (1.5) Two 2-complexes are (up to 2-deformation) spines of a dual decomposition of the 4-sphere if and only if they satisfy the conclusions of Alexander-Lefshetz duality ( and ). (3.3) If is 1-connected then there is a ``pseudo'' handle decomposition without 1-handles, in the sense that there is a pseudo collar (a relative 2-handlebody with spine that 2-deforms to ) and is obtained from this by attaching handles of index .

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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/S0002-9947-01-02940-3

Received by editor(s):
October 2, 2000

Received by editor(s) in revised form:
August 4, 2001

Published electronically:
November 8, 2001

Article copyright:
© Copyright 2001
American Mathematical Society