Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2005; 42(4): 773-793

Printed July 1, 2005

Copyright © The Korean Mathematical Society.

Disjoint pairs of annuli and disks for Heegaard splittings

Toshio Saito

Osaka University

Abstract

We consider interesting conditions, one of which will be called the disjoint $(A^2,D^2)$-pair property, on genus $g\ge 2$ Heegaard splittings of compact orientable $3$-manifolds. Here a Heegaard splitting $(C_1,C_2;F)$ admits the disjoint $(A^2,D^2)$-pair property if there are an essential annulus $A_i$ normally embedded in $C_i$ and an essential disk $D_j$ in $C_j$ ($(i,j)=(1,2)$ or $(2,1)$) such that $\partial A_i$ is disjoint from $\partial D_j$. It is proved that all genus $g\ge 2$ Heegaard splittings of toroidal manifolds and Seifert fibered spaces admit the disjoint $(A^2,D^2)$-pair property.

Keywords: Heegaard splittings, the disjoint $(A^2,D^2)$-pair property.

MSC numbers: 57N10

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