Geometry & Topology

Genus-two trisections are standard

Jeffrey Meier and Alexander Zupan

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We show that the only closed 4–manifolds admitting genus-two trisections are S2 × S2 and connected sums of S1 × S3, 2 and ¯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 S3. As a corollary, we classify tunnel number one links with an integral cosmetic Dehn surgery.

Article information

Geom. Topol., Volume 21, Number 3 (2017), 1583-1630.

Received: 1 December 2014
Revised: 21 February 2016
Accepted: 25 March 2016
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N12: Topology of $E^3$ and $S^3$ [See also 57M40] 57R65: Surgery and handlebodies
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

trisections Heegaard splittings waves


Meier, Jeffrey; Zupan, Alexander. Genus-two trisections are standard. Geom. Topol. 21 (2017), no. 3, 1583--1630. doi:10.2140/gt.2017.21.1583.

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  • J Berge, Embedding the exterior of one-tunnel knots and links in the $3$–sphere, unpublished manuscript
  • B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton University Press (2012)
  • M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357–453
  • M H Freedman, F Quinn, Topology of $4$–manifolds, Princeton Mathematical Series 39, Princeton University Press (1990)
  • D Gabai, Foliations and surgery on knots, Bull. Amer. Math. Soc. 15 (1986) 83–87
  • D Gay, R Kirby, Trisecting $4$–manifolds, Geom. Topol. 20 (2016) 3097–3132
  • R E Gompf, M Scharlemann, A Thompson, Fibered knots and potential counterexamples to the property 2R and slice-ribbon conjectures, Geom. Topol. 14 (2010) 2305–2347
  • R E Gompf, A I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Mathematics 20, Amer. Math. Soc., Providence, RI (1999)
  • A Haas, P Susskind, The geometry of the hyperelliptic involution in genus two, Proc. Amer. Math. Soc. 105 (1989) 159–165
  • W Haken, Some results on surfaces in $`3`$–manifolds, from “Studies in Modern Topology” (P J Hilton, editor), Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, NJ) (1968) 39–98
  • T Homma, M Ochiai, M-o Takahashi, An algorithm for recognizing $S\sp{3}$ in $3$–manifolds with Heegaard splittings of genus two, Osaka J. Math. 17 (1980) 625–648
  • K Ishihara, On tunnel number one links with surgeries yielding the $3$–sphere, Osaka J. Math. 47 (2010) 189–208
  • K Johannson, Topology and combinatorics of $3$–manifolds, Lecture Notes in Mathematics 1599, Springer, New York (1995)
  • F Laudenbach, V Poénaru, A note on $4$–dimensional handlebodies, Bull. Soc. Math. France 100 (1972) 337–344
  • H Matsuda, M Ozawa, K Shimokawa, On non-simple reflexive links, J. Knot Theory Ramifications 11 (2002) 787–791
  • E Mayrand, Dehn fillings on a two torus boundary components $3$–manifold, Osaka J. Math. 39 (2002) 779–793
  • J Meier, T Schirmer, A Zupan, Classification of trisections and the generalized property R conjecture, Proc. Amer. Math. Soc. 144 (2016) 4983–4997
  • J Meier, A Zupan, Bridge trisections of knotted surfaces in $S^4$, preprint (2015)
  • M Ochiai, Heegaard diagrams and Whitehead graphs, Math. Sem. Notes Kobe Univ. 7 (1979) 573–591
  • M Ochiai, Heegaard diagrams of $3$–manifolds, Trans. Amer. Math. Soc. 328 (1991) 863–879
  • M Teragaito, Links with surgery yielding the $3$–sphere, J. Knot Theory Ramifications 11 (2002) 105–108