TY - JOUR

T1 - Haken spheres for genus two Heegaard splittings

AU - Cho, Sangbum

AU - Koda, Yuya

N1 - Publisher Copyright:
Copyright © Cambridge Philosophical Society 2017.
Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

PY - 2018/11/1

Y1 - 2018/11/1

N2 - A manifold which admits a reducible genus-2 Heegaard splitting is one of the 3-sphere, S2 × S1, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is the 3-sphere, S2 × S1 or a connected sum whose summands are lens spaces or S2 × S1, the combinatorial structure of the complex has been studied by several authors. In particular, it was shown that those complexes are all contractible. In this work, we study the remaining cases, that is, when the manifolds are lens spaces. We give a precise description of each of the complexes for the genus-2 Heegaard splittings of lens spaces. A remarkable fact is that the complexes for most lens spaces are not contractible and even not connected.

AB - A manifold which admits a reducible genus-2 Heegaard splitting is one of the 3-sphere, S2 × S1, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is the 3-sphere, S2 × S1 or a connected sum whose summands are lens spaces or S2 × S1, the combinatorial structure of the complex has been studied by several authors. In particular, it was shown that those complexes are all contractible. In this work, we study the remaining cases, that is, when the manifolds are lens spaces. We give a precise description of each of the complexes for the genus-2 Heegaard splittings of lens spaces. A remarkable fact is that the complexes for most lens spaces are not contractible and even not connected.

UR - http://www.scopus.com/inward/record.url?scp=85032201456&partnerID=8YFLogxK

U2 - 10.1017/S0305004117000718

DO - 10.1017/S0305004117000718

M3 - Article

AN - SCOPUS:85032201456

VL - 165

SP - 563

EP - 572

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 3

ER -