Sweepouts of amalgamated 3–manifolds

Bachman, David; Schleimer, Saul; Sedgwick, Eric

doi:10.2140/agt.2006.6.171

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Sweepouts of amalgamated 3–manifolds

David Bachman, Saul Schleimer and Eric Sedgwick

Algebraic & Geometric Topology 6 (2006) 171–194

DOI: 10.2140/agt.2006.6.171

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 $X \cup_{F} Y$ 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 \cup_{F} Y) \geq \frac{1}{2} (g (X) + g (Y) - 2 g (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

Volume 6, issue 1 (2006)