Bulletin of the
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ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2019; 56(2): 515-520

Online first article October 29, 2018      Printed March 1, 2019

https://doi.org/10.4134/BKMS.b180383

Copyright © The Korean Mathematical Society.

Bicompressible surfaces and incompressible surfaces

Toshio Saito

Joetsu University of Education

Abstract

We give new evidence that ``complicated'' Heegaard surfaces behave like incompressible surfaces. More precisely, suppose that a closed connected orientable 3-manifold $M$ contains a closed connected incompressible surface $F$ which separates $M$ into two (connected) components $M_1$ and $M_2$. Let $S$ be a Heegaard surface of $M$. Our result is that if the Hempel distance of $S$ is at least four, then $S$ is isotoped so that $S\cap M_i$ is incompressible for each $i=1,2$.

Keywords: bicompressible surface, incompressible surface, Heegaard surface, Hempel distance

MSC numbers: Primary 57N10

Supported by: This work is supported by JSPS KAKENHI Grant Number 15K04869

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