Volume 6, issue 1 (2006)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Sweepouts of amalgamated 3–manifolds

David Bachman, Saul Schleimer and Eric Sedgwick

Algebraic & Geometric Topology 6 (2006) 171–194

arXiv: math.GT/0507490

Abstract

We show that if two 3–manifolds with toroidal boundary are glued via a “sufficiently complicated" map then every Heegaard splitting of the resulting 3–manifold is weakly reducible. Additionally, suppose X FY is a manifold obtained by gluing X and Y , two connected small manifolds with incompressible boundary, along a closed surface F. Then the following inequality on genera is obtained:

g(X FY ) 1 2 g(X) + g(Y ) 2g(F).

Both results follow from a new technique to simplify the intersection between an incompressible surface and a strongly irreducible Heegaard splitting.

Keywords
Heegaard splitting, incompressible surface
Mathematical Subject Classification 2000
Primary: 57N10, 57M99
Secondary: 57M27
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Publication
Received: 26 July 2005
Revised: 18 January 2006
Accepted: 26 January 2006
Published: 24 February 2006
Authors
David Bachman
Mathematics Department
Pitzer College
1050 North Mills Avenue
Claremont CA 91711
USA
Saul Schleimer
Department of Mathematics
Rutgers
The State University of New Jersey
110 Frelinghuysen Rd
Piscataway NJ 08854-8019
USA
Eric Sedgwick
CTI
DePaul University
243 S Wabash Avenue
Chicago IL 60604
USA