TY - JOUR

T1 - Connected primitive disk complexes and genus two goeritz groups of lens spaces

AU - Cho, Sangbum

AU - Koda, Yuya

N1 - Funding Information:
This work was supported in part by Basic Science Research Program through the National Research Foundation of Korea [NRF-2015R1A1A1A05001071] funded by the Ministry of Science, ICT(Information & Communication Technology) and Future Planning to S. C.; and Grant-in-Aid for Young Scientists (B) [No. 26800028], Japan Society for the Promotion of Science to Y. K.
Publisher Copyright:
© The Author(s) 2016.

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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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U2 - 10.1093/imrn/rnv399

DO - 10.1093/imrn/rnv399

M3 - Article

AN - SCOPUS:85016233562

SN - 1073-7928

VL - 2016

SP - 7302

EP - 7340

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 23

ER -