A Strong Haken's Theorem

Martin Scharlemann

doi:10.2140/agt.2024.24.717

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arXiv · 2003.08523

A Strong Haken's Theorem

Martin Scharlemann

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Abstract

Suppose T is a Heegaard splitting surface for a compact orientable 3-manifold M, and S is a reducing sphere for M. In 1968 Haken showed that there is then also a reducing sphere S* for the Heegaard splitting. That is, S* is a reducing sphere for M and the surfaces T and S* intersect in a single circle. In 1987 Casson and Gordon extended the result to boundary-reducing disks in M and noted that in both cases S* is obtained from S by a sequence of operations called 1-surgeries. Here we show that in fact one may take S* = S.

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Martin Scharlemann. 2020-03-19. A Strong Haken's Theorem. https://doi.org/10.2140/agt.2024.24.717

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