Heegaard splittings of branched coverings of 𝑆³

Birman, Joan S.; Hilden, Hugh M.

doi:10.1090/S0002-9947-1975-0380765-8

Heegaard splittings of branched coverings of $S^{3}$
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by Joan S. Birman and Hugh M. Hilden

Trans. Amer. Math. Soc. 213 (1975), 315-352

DOI: https://doi.org/10.1090/S0002-9947-1975-0380765-8

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Abstract:

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 We will give a counterexample to a conjecture (unpublished) of W. Haken, which would have implied that ${S^3}$ could be identified (in the class of all simply-connected 3-manifolds) by the property that certain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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