Heegaard surfaces and the distance of amalgamation

Li, Tao

doi:10.2140/gt.2010.14.1871

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Heegaard surfaces and the distance of amalgamation

Tao Li

Geometry & Topology 14 (2010) 1871–1919

DOI: 10.2140/gt.2010.14.1871

Abstract

Let $M_{1}$ and $M_{2}$ be orientable irreducible $3$ –manifolds with connected boundary and suppose $\partial M_{1} ≅ \partial M_{2}$ . Let $M$ be a closed $3$ –manifold obtained by gluing $M_{1}$ to $M_{2}$ along the boundary. We show that if the gluing homeomorphism is sufficiently complicated, then $M$ is not homeomorphic to $S^{3}$ and all small-genus Heegaard splittings of $M$ are standard in a certain sense. In particular, $g (M) = g (M_{1}) + g (M_{2}) - g (\partial M_{i})$ , 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 2000

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/

Volume 14, issue 4 (2010)