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Haken spheres for genus two Heegaard splittings

Published online by Cambridge University Press:  18 October 2017

SANGBUM CHO  and
YUYA KODA
SANGBUM CHO
Affiliation:
Department of Mathematics Education, Hanyang University, Seoul 04763, Korea. e-mail: scho@hanyang.ac.kr
YUYA KODA
Affiliation:
Department of Mathematics, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526, Japan. e-mail: ykoda@hiroshima-u.ac.jp
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Abstract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 3 , November 2018 , pp. 563 - 572
Copyright
Copyright © Cambridge Philosophical Society 2017 

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