A topologically minimal, weakly reducible, unstabilized Heegaard splitting of genus three is critical

Kim, Jungsoo

doi:10.2140/agt.2016.16.1427

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A topologically minimal, weakly reducible, unstabilized Heegaard splitting of genus three is critical

Jungsoo Kim

Algebraic & Geometric Topology 16 (2016) 1427–1451

DOI: 10.2140/agt.2016.16.1427

Abstract

Let $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo class="MathClass-open">(</mo><mrow><mi mathvariant="bold-script">V</mi><mo class="MathClass-punc">,</mo><mi mathvariant="bold-script">W</mi><mo class="MathClass-punc">;</mo><mi>F</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ be a weakly reducible unstabilized genus three Heegaard splitting in an orientable, irreducible $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn></math>$ –manifold $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi><mspace width="0.3em" class="thinspace"/></math>$ . In this article, we prove that either the disk complex $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="bold-script">D</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>F</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ is contractible or $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>F</mi></math>$ is critical. Hence, the topological index of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>F</mi></math>$ is $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn></math>$ if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>F</mi></math>$ is topologically minimal.

Keywords

$3$ –manifold, Heegaard splitting, disk complex, topologically minimal surface

Mathematical Subject Classification 2010

Primary: 57M50

References

Publication

Received: 18 August 2014

Revised: 25 July 2015

Accepted: 19 September 2015

Published: 1 July 2016

Authors

Jungsoo Kim
BK21 PLUS SNU Mathematical Sciences Division Seoul National University 1 Gwanak-ro Gwanak-Gu Seoul 08826 South Korea

Volume 16, issue 3 (2016)