Connected Primitive Disk Complexes and Genus Two Goeritz Groups of Lens Spacesopen access
- Authors
- Cho, Sangbum; Koda, Yuya
- Issue Date
- Dec-2016
- Publisher
- OXFORD UNIV PRESS
- Citation
- INTERNATIONAL MATHEMATICS RESEARCH NOTICES, v.2016, no.23, pp.7302 - 7340
- Indexed
- SCIE
SCOPUS
- Journal Title
- INTERNATIONAL MATHEMATICS RESEARCH NOTICES
- Volume
- 2016
- Number
- 23
- Start Page
- 7302
- End Page
- 7340
- URI
- https://scholarworks.bwise.kr/hanyang/handle/2021.sw.hanyang/153453
- DOI
- 10.1093/imrn/rnv399
- ISSN
- 1073-7928
- Abstract
- Given a stabilized Heegaard splitting of a three-manifold, the primitive disk complex for the splitting is the subcomplex of the disk complex for a handlebody in the splitting spanned by the vertices of the primitive disks. In this work, we study the structure of the primitive disk complex for the genus-2 Heegaard splitting of each lens space. In particular, we show that the complex for the genus-2 splitting for the lens space L(p, q) with 1 ≤ q ≤ p/2 is connected if and only if p ≡ ±1 (mod q), and describe the combinatorial structure of each of those complexes. As an application, we obtain a finite presentation of the genus-2 Goeritz group of each of those lens spaces, the group of isotopy classes of orientation preserving homeomorphisms of the lens space that preserve the genus-2 Heegaard splitting of it.
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