Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Classification of trisections and the Generalized Property R Conjecture


Authors: Jeffrey Meier, Trent Schirmer and Alexander Zupan
Journal: Proc. Amer. Math. Soc. 144 (2016), 4983-4997
MSC (2010): Primary 57M25, 57M99, 57Q25
DOI: https://doi.org/10.1090/proc/13105
Published electronically: May 24, 2016
MathSciNet review: 3544545
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the members of a large class of unbalanced four-manifold trisections are standard, and we present a family of trisections that is likely to include non-standard trisections of the four-sphere. As an application, we prove a stable version of the Generalized Property R Conjecture for $ c$-component links with tunnel number at most $ c$.


References [Enhancements On Off] (What's this?)

  • [1] J. J. Andrews and M. L. Curtis, Free groups and handlebodies, Proc. Amer. Math. Soc. 16 (1965), 192-195. MR 0173241
  • [2] Leonardo N. Carvalho and Ulrich Oertel, A classification of automorphisms of compact 3-manifolds, arXiv:0510610, 2005.
  • [3] Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990.
  • [4] Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no. 3, 357-453. MR 679066
  • [5] David Gabai, Foliations and the topology of $ 3$-manifolds. II, J. Differential Geom. 26 (1987), no. 3, 461-478. MR 910017
  • [6] David Gabai, Foliations and the topology of $ 3$-manifolds. III, J. Differential Geom. 26 (1987), no. 3, 479-536. MR 910018
  • [7] David Gay and Robion Kirby, Trisecting 4-manifolds, arXiv:1205.1565v3, 2012.
  • [8] Robert E. Gompf, Martin Scharlemann, and Abigail Thompson, Fibered knots and potential counterexamples to the property 2R and slice-ribbon conjectures, Geom. Topol. 14 (2010), no. 4, 2305-2347. MR 2740649, https://doi.org/10.2140/gt.2010.14.2305
  • [9] Robert E. Gompf and András I. Stipsicz, $ 4$-manifolds and Kirby calculus, Graduate Studies in Mathematics, vol. 20, American Mathematical Society, Providence, RI, 1999.
  • [10] C. McA. Gordon, Combinatorial methods in Dehn surgery, Lectures at KNOTS '96 (Tokyo), Ser. Knots Everything, vol. 15, World Sci. Publ., River Edge, NJ, 1997, pp. 263-290. MR 1474525, https://doi.org/10.1142/9789812796097_0010
  • [11] Wolfgang Haken, Some results on surfaces in $ 3$-manifolds, Studies in Modern Topology, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.), 1968, pp. 39-98. MR 0224071
  • [12] Klaus Johannson, Topology and combinatorics of 3-manifolds, Lecture Notes in Mathematics, vol. 1599, Springer-Verlag, Berlin, 1995. MR 1439249
  • [13] F. Laudenbach, On the $ 2$-spheres in a $ 3$-manifold, Bull. Amer. Math. Soc. 78 (1972), 792-795. MR 0303554
  • [14] François Laudenbach and Valentin Poénaru, A note on $ 4$-dimensional handlebodies, Bull. Soc. Math. France 100 (1972), 337-344. MR 0317343
  • [15] Fengchun Lei, On stability of Heegaard splittings, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 1, 55-57. MR 1757777, https://doi.org/10.1017/S0305004100004461
  • [16] Jeffrey Meier and Alexander Zupan, Genus two trisections are standard, arXiv:1410.8133, 2014.
  • [17] Peter S. Ozsváth and Zoltán Szabó, Knot Floer homology and integer surgeries, Algebr. Geom. Topol. 8 (2008), no. 1, 101-153. MR 2377279, https://doi.org/10.2140/agt.2008.8.101
  • [18] Peter Ozsváth and Zoltán Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. (2) 159 (2004), no. 3, 1159-1245. MR 2113020, https://doi.org/10.4007/annals.2004.159.1159
  • [19] Kurt Reidemeister, Zur dreidimensionalen Topologie, Abh. Math. Sem. Univ. Hamburg 9 (1933), no. 1, 189-194 (German). MR 3069596, https://doi.org/10.1007/BF02940644
  • [20] Martin Scharlemann, Producing reducible $ 3$-manifolds by surgery on a knot, Topology 29 (1990), no. 4, 481-500. MR 1071370, https://doi.org/10.1016/0040-9383(90)90017-E
  • [21] Martin Scharlemann, Heegaard splittings of compact 3-manifolds, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 921-953. MR 1886684
  • [22] Saul Schleimer, Waldhausen's theorem, Workshop on Heegaard Splittings, Geom. Topol. Monogr., vol. 12, Geom. Topol. Publ., Coventry, 2007, pp. 299-317. MR 2408252, https://doi.org/10.2140/gtm.2007.12.299
  • [23] James Singer, Three-dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc. 35 (1933), no. 1, 88-111. MR 1501673, https://doi.org/10.2307/1989314
  • [24] Friedhelm Waldhausen, Heegaard-Zerlegungen der $ 3$-Sphäre, Topology 7 (1968), 195-203 (German). MR 0227992
  • [25] Michael J. Williams, Handle number one links and generalized property $ R$, Proc. Amer. Math. Soc. 140 (2012), no. 3, 1105-1109. MR 2869095, https://doi.org/10.1090/S0002-9939-2011-10966-3

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 57M25, 57M99, 57Q25

Retrieve articles in all journals with MSC (2010): 57M25, 57M99, 57Q25


Additional Information

Jeffrey Meier
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47408
Email: jlmeier@indiana.edu

Trent Schirmer
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
Email: trent.schirmer@okstate.edu

Alexander Zupan
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
Address at time of publication: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska, 68588
Email: zupan@unl.edu

DOI: https://doi.org/10.1090/proc/13105
Received by editor(s): March 30, 2015
Received by editor(s) in revised form: August 6, 2015, and January 14, 2016
Published electronically: May 24, 2016
Communicated by: Martin Scharlemann
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society