A Strong Haken's Theorem
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866910418852642816 |
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| author | Scharlemann, Martin |
| author_facet | Scharlemann, Martin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2003_08523 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A Strong Haken's Theorem Scharlemann, Martin Geometric Topology 57M50 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. |
| title | A Strong Haken's Theorem |
| topic | Geometric Topology 57M50 |
| url | https://arxiv.org/abs/2003.08523 |