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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060

Heegaard surfaces and the distance of amalgamation

Tao Li

Geometry & Topology 14 (2010) 1871–1919

DOI: 10.2140/gt.2010.14.1871

Abstract

Let M1 and M2 be orientable irreducible 3–manifolds with connected boundary and suppose ∂M1∂M2. Let M be a closed 3–manifold obtained by gluing M1 to M2 along the boundary. We show that if the gluing homeomorphism is sufficiently complicated, then M is not homeomorphic to S3 and all small-genus Heegaard splittings of M are standard in a certain sense. In particular, g(M) = g(M1) + g(M2) g(∂Mi), where g(M) denotes the Heegaard genus of M. This theorem is also true for certain manifolds with multiple boundary components.

Keywords

Heegaard splitting, amalgamation, curve complex

Mathematical Subject Classification

Primary: 57N10

Secondary: 57M50

References
Publication

Received: 31 July 2008
Revised: 9 March 2010
Accepted: 7 June 2010
Published: 21 July 2010
Proposed: Cameron Gordon
Seconded: Joan Birman, Colin Rourke

Authors
Tao Li
Department of Mathematics
Boston College
Chestnut Hill, MA 02467
http://www2.bc.edu/~taoli/