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The Heegaard structure of Dehn filled manifolds

Yoav Moriah and Eric Sedgwick

Geometry & Topology Monographs 12 (2007) 233–263

DOI: 10.2140/gtm.2007.12.233

arXiv: 0706.1927

Abstract

We expect manifolds obtained by Dehn filling to inherit properties from the knot manifold. To what extent does that hold true for the Heegaard structure? We study four changes to the Heegaard structure that may occur after filling: (1) Heegaard genus decreases, (2) a new Heegaard surface is created, (3) a non-stabilized Heegaard surface destabilizes, and (4) two or more non-isotopic Heegaard surfaces become isotopic. We survey general results that give quite satisfactory restrictions to phenomena (1) and (2) and, in a parallel thread, give a complete classification of when all four phenomena occur when filling most torus knot exteriors. This latter thread yields sufficient (and perhaps necessary) conditions for the occurrence of phenomena (3) and (4).

Keywords

Heegaard splitting, Dehn surgery, Dehn filling, torus knot, Seifert fibered space

Mathematical Subject Classification

Primary: 57N10

Secondary: 57M27

References
Publication

Received: 23 May 2007
Revised: 3 July 2007
Accepted: 3 July 2007
Published: 3 December 2007

Authors
Yoav Moriah
Department of Mathematics
Technion – Israel Institute of Technology
Haifa 32000
Israel
Eric Sedgwick
DePaul CTI
243 S Wabash Avenue
Chicago IL 60604
USA