Arc complexes, sphere complexes and Goeritz groups

Cho, Sangbum; Koda, Yuya; Seo, Arim

doi:10.48550/arxiv.1403.7832

Cited by 2 publications

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“…For example, from the works [10], [18], [1] and [4], a finite presentation of the genus-2 Goeritz group of the 3-sphere was obtained and from [6], that of S 2 × S 1 was obtained. We refer the reader to [12], [13], [19], [7], [8] for finite presentations or finite generating sets of the Goeritz groups of several Heegaard splittings. For the genus-2 Goeriz groups of lens spaces, the finite presentations are obtained only for the lens spaces L(p, 1) in [5].…”

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confidence: 99%

Connected Primitive Disk Complexes and Genus Two Goeritz Groups of Lens Spaces

Cho

Koda

2016

Int Math Res Notices

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Given a stabilized Heegaard splitting of a 3-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 two Heegaard splitting of each lens space. In particular, we show that the complex for the genus two 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 two 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 two Heegaard splitting of it.

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