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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Dual decompositions of 4-manifolds
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by Frank Quinn PDF
Trans. Amer. Math. Soc. 354 (2002), 1373-1392 Request permission

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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Additional Information
  • Frank Quinn
  • Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061-0123
  • Email: quinn@math.vt.edu
  • Received by editor(s): October 2, 2000
  • Received by editor(s) in revised form: August 4, 2001
  • Published electronically: November 8, 2001
  • © Copyright 2001 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 1373-1392
  • MSC (2000): Primary 57R65, 57M20
  • DOI: https://doi.org/10.1090/S0002-9947-01-02940-3
  • MathSciNet review: 1873010