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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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Heegaard splittings of branched coverings of $S^{3}$
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by Joan S. Birman and Hugh M. Hilden PDF
Trans. Amer. Math. Soc. 213 (1975), 315-352 Request permission


This paper concerns itself with the relationship between two seemingly different methods for representing a closed, orientable 3-manifold: on the one hand as a Heegaard splitting, and on the other hand as a branched covering of the 3-sphere. The ability to pass back and forth between these two representations will be applied in several different ways: 1. It will be established that there is an effective algorithm to decide whether a 3-manifold of Heegaard genus 2 is a 3-sphere. 2. We will show that the natural map from 6-plat representations of knots and links to genus 2 closed oriented 3-manifolds is injective and surjective. This relates the question of whether or not Heegaard splittings of closed, oriented 3-manifolds are “unique” to the question of whether plat representations of knots and links are “unique". 3. We will give a counterexample to a conjecture (unpublished) of W. Haken, which would have implied that ${S^3}$ could be identified (in the class of all simply-connected 3-manifolds) by the property that certain canonical presentations for ${\pi _1}{S^3}$ are always “nice". The final section of the paper studies a special class of genus 2 Heegaard splittings: the 2-fold covers of ${S^3}$ which are branched over closed 3-braids. It is established that no counterexamples to the “genus 2 Poincaré conjecture” occur in this class of 3-manifolds.
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 213 (1975), 315-352
  • MSC: Primary 55A10
  • DOI:
  • MathSciNet review: 0380765