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Genus one open books with non-left-orderable fundamental group


Authors: Yu Li and Liam Watson
Journal: Proc. Amer. Math. Soc. 142 (2014), 1425-1435
MSC (2010): Primary 57M05, 57M12, 20F60, 06F15
DOI: https://doi.org/10.1090/S0002-9939-2014-11847-8
Published electronically: January 10, 2014
MathSciNet review: 3162262
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ Y$ be a closed, connected, orientable three-manifold admitting a genus one open book decomposition with one boundary component. We prove that if $ Y$ is an L-space, then the fundamental group of $ Y$ is not left-orderable. This answers a question posed by John Baldwin.


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

  • [1] John A. Baldwin, Heegaard Floer homology and genus one, one-boundary component open books, J. Topol. 1 (2008), no. 4, 963-992. MR 2461862 (2010e:57012), https://doi.org/10.1112/jtopol/jtn029
  • [2] Steven Boyer, Cameron McA. Gordon, and Liam Watson, On L-spaces and left-orderable fundamental groups, Math. Ann. 356 (2013), no. 4, 1213-1245. MR 3072799
  • [3] Steven Boyer, Dale Rolfsen, and Bert Wiest, Orderable 3-manifold groups, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 243-288 (English, with English and French summaries). MR 2141698 (2006a:57001)
  • [4] Adam Clay and Liam Watson, Left-orderable fundamental groups and Dehn surgery, Int. Math. Res. Not. IMRN 2013, no. 12, 2862-2890. MR 3071667
  • [5] Adam Clay and Liam Watson, On cabled knots, Dehn surgery, and left-orderable fundamental groups, Math. Res. Lett. 18 (2011), no. 6, 1085-1095. MR 2915469
  • [6] Joshua E. Greene, A spanning tree model for the Heegaard Floer homology of a branched double-cover, J. Topol. 6 (2013), no. 2, 525-567. MR 3065184
  • [7] -, Alternating links and left-orderability, preprint, arXiv:1107.5232.
  • [8] Tetsuya Ito, Non-left-orderable double branched coverings, Algebr. Geom. Topol. 13 (2013), no. 4, 1937-1965. MR 3073904
  • [9] Kunio Murasugi, On closed $ 3$-braids, Memoirs of the American Mathmatical Society, No. 151. American Mathematical Society, Providence, RI, 1974. MR 0356023 (50 #8496)
  • [10] Peter Ozsváth and Zoltán Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005), no. 6, 1281-1300. MR 2168576 (2006f:57034), https://doi.org/10.1016/j.top.2005.05.001
  • [11] Peter Ozsváth and Zoltán Szabó, On the Heegaard Floer homology of branched double-covers, Adv. Math. 194 (2005), no. 1, 1-33. MR 2141852 (2006e:57041), https://doi.org/10.1016/j.aim.2004.05.008
  • [12] Thomas Peters, On L-spaces and non left-orderable 3-manifold groups, preprint, arXiv:0903.4495.
  • [13] Peter Scott, The geometries of $ 3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401-487. MR 705527 (84m:57009), https://doi.org/10.1112/blms/15.5.401
  • [14] Liam Watson, A surgical perspective on quasi-alternating links, Low-dimensional and symplectic topology, Proc. Sympos. Pure Math., vol. 82, Amer. Math. Soc., Providence, RI, 2011, pp. 39-51. MR 2768652

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Additional Information

Yu Li
Affiliation: School of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China
Address at time of publication: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706-1325
Email: adterram@math.wisc.edu

Liam Watson
Affiliation: Department of Mathematics, University of California Los Angeles, 520 Portola Plaza, Los Angeles, California 90095
Address at time of publication: School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, Glasgow G12 8QW, United Kingdom
Email: Liam.Watson@glasgow.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-2014-11847-8
Received by editor(s): October 11, 2011
Received by editor(s) in revised form: April 10, 2012, and May 1, 2012
Published electronically: January 10, 2014
Additional Notes: The first author was supported by a CSST undergraduate summer internship
The second author was partially supported by an NSERC postdoctoral fellowship
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2014 American Mathematical Society

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