A strong Haken theorem

Scharlemann, Martin

doi:10.2140/agt.2024.24.717

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A strong Haken theorem

Martin Scharlemann

Algebraic & Geometric Topology 24 (2024) 717–753

DOI: 10.2140/agt.2024.24.717

Abstract

Suppose $M = A \cup_{T} B$ is a Heegaard split compact orientable $3$ –manifold and $S \subset M$ is a reducing sphere for $M$ . Haken (1968) showed that there is then also a reducing sphere $S^{*}$ for the Heegaard splitting. Casson and Gordon (1987) extended the result to $\partial$ –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 $S^{*} = S$ .

Keywords

Heegaard splitting, compression body, reducing disks and spheres

Mathematical Subject Classification

Primary: 57K35

References

Publication

Received: 8 April 2020

Revised: 1 August 2022

Accepted: 16 August 2022

Published: 12 April 2024

Authors

Martin Scharlemann
Department of Mathematics University of California, Santa Barbara Santa Barbara, CA United States

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Volume 24, issue 2 (2024)