Genus-two trisections are standard

Meier, Jeffrey; Zupan, Alexander

doi:10.2140/gt.2017.21.1583

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Genus-two trisections are standard

Jeffrey Meier and Alexander Zupan

Geometry & Topology 21 (2017) 1583–1630

DOI: 10.2140/gt.2017.21.1583

Abstract

We show that the only closed $4$ –manifolds admitting genus-two trisections are $S^{2} \times S^{2}$ and connected sums of $S^{1} \times S^{3}$ , ${ℂ ℙ}^{2}$ and ${\bar{ℂ ℙ}}^{2}$ with two summands. Moreover, each of these manifolds admits a unique genus-two trisection up to diffeomorphism. The proof relies heavily on the combinatorics of genus-two Heegaard diagrams of $S^{3}$ . As a corollary, we classify tunnel number one links with an integral cosmetic Dehn surgery.

Keywords

trisections, Heegaard splittings, waves

Mathematical Subject Classification 2010

Primary: 57N12, 57R65

Secondary: 57M25

References

Publication

Received: 1 December 2014

Revised: 21 February 2016

Accepted: 25 March 2016

Published: 10 May 2017

Proposed: Colin Rourke

Seconded: David Gabai, Ronald Stern

Authors

Jeffrey Meier
Department of Mathematics Indiana University Rawles Hall 425 831 East 3rd Street Bloomington, IN 47405 United States

Alexander Zupan
Department of Mathematics University of Texas at Austin 1 University Station C1200 Austin, TX 78712 United States

Volume 21, issue 3 (2017)