A Strong Haken's Theorem
Geometric Topology
2024-04-24 v5
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.
Cite
@article{arxiv.2003.08523,
title = {A Strong Haken's Theorem},
author = {Martin Scharlemann},
journal= {arXiv preprint arXiv:2003.08523},
year = {2024}
}
Comments
Final version, to appear in Algebraic and Geometric Topology