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A strong Haken theorem

Martin Scharlemann

Algebraic & Geometric Topology 24 (2024) 717–753
Abstract

Suppose M = A TB is a Heegaard split compact orientable 3–manifold and S 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 –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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