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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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 $\leq 2$. In dimensions $\geq 5$ results of Smale (trivial $\pi _{1}$) and Wall (general $\pi _{1}$) 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 ($H_{1}K\simeq H^{2}L$ and $H_{2}K\simeq H^{1}L$). (3.3) If $(N,\partial N)$ is 1-connected then there is a “pseudo” handle decomposition without 1-handles, in the sense that there is a pseudo collar $(M,\partial N)$ (a relative 2-handlebody with spine that 2-deforms to $\partial N$) and $N$ is obtained from this by attaching handles of index $\geq 2$.

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Frank Quinn
Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061-0123

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