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L-spaces, taut foliations, and graph manifolds

Part of: Low-dimensional topology

Published online by Cambridge University Press:  23 January 2020

Jonathan Hanselman ,
Jacob Rasmussen ,
Sarah Dean Rasmussen  and
Liam Watson
Jonathan Hanselman
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road,Princeton, NJ08540, USA email jh66@math.princeton.edu
Jacob Rasmussen
Affiliation:
Department of Pure Mathematics and Mathematical Statistics,Centre for Mathematical Sciences, Wilberforce Road, CambridgeCB3 0WB, UK email J.Rasmussen@dpmms.cam.ac.uk
Sarah Dean Rasmussen
Affiliation:
Department of Pure Mathematics and Mathematical Statistics,Centre for Mathematical Sciences, Wilberforce Road, CambridgeCB3 0WB, UK email S.Rasmussen@dpmms.cam.ac.uk
Liam Watson
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, CanadaV6T 1Z2 email liam@math.ubc.ca
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Abstract

If $Y$ is a closed orientable graph manifold, we show that $Y$ admits a coorientable taut foliation if and only if $Y$ is not an L-space. Combined with previous work of Boyer and Clay, this implies that $Y$ is an L-space if and only if $\unicode[STIX]{x1D70B}_{1}(Y)$ is not left-orderable.

Keywords

graph manifoldL-spacetaut foliationHeegaard Floer

MSC classification

Primary: 57M27: Invariants of knots and 3-manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 3 , March 2020 , pp. 604 - 612
Copyright
© The Authors 2020

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Footnotes

The first author was partially supported by NSF RTG grant DMS-1148490. The second author was partially supported by EPSRC grant EP/M000648/1. The third author was supported by EPSRC grant EP/M000648/1. The fourth author was partially supported by a Marie Curie Career Integration Grant (HFFUNDGRP).

References

Bowden, J., Approximating C 0-foliations by contact structures, Geom. Funct. Anal. 26 (2016), 1255–1296.10.1007/s00039-016-0387-2CrossRefGoogle Scholar
Boyer, S. and Clay, A., Slope detection, foliations in graph manifolds, and L-spaces, Preprint (2015), arXiv:1510.02378.Google Scholar
Boyer, S. and Clay, A., Foliations, orders, representations, L-spaces and graph manifolds, Adv. Math. 310 (2017), 159–234.10.1016/j.aim.2017.01.026CrossRefGoogle Scholar
Boyer, S., Gordon, C. McA. and Watson, L., On L-spaces and left-orderable fundamental groups, Math. Ann. 356 (2013), 1213–1245.10.1007/s00208-012-0852-7CrossRefGoogle Scholar
Boyer, S., Rolfsen, D. and Wiest, B., Orderable 3-manifold groups, Ann. Inst. Fourier (Grenoble) 55 (2005), 243–288.10.5802/aif.2098CrossRefGoogle Scholar
Gabai, D., Foliations and the topology of 3-manifolds, J. Differential Geom. 18 (1983), 445–503.10.4310/jdg/1214437784CrossRefGoogle Scholar
Gillespie, T., L-space fillings and generalized solid tori, Preprint (2016), arXiv:1603.05016.Google Scholar
Hanselman, J., Bordered Heegaard Floer homology and graph manifolds, Algebr. Geom. Topol. 16 (2016), 3103–3166.10.2140/agt.2016.16.3103CrossRefGoogle Scholar
Hanselman, J. and Watson, L., A calculus for bordered Floer homology, Preprint (2015),arXiv:1508.05445.Google Scholar
Kazez, W. H. and Roberts, R., Approximating C 1, 0-foliations, in Interactions between low-dimensional topology and mapping class groups, Geometry & Topology Monographs, vol. 19, eds Baykur, R. I., Etnyre, J. and Hamenstädt, U. (Mathematical Sciences Publishers, Berkeley, CA, 2015), 21–72.Google Scholar
Kazez, W. H. and Roberts, R., C 0 approximations of foliations, Geom. Topol. 21 (2017), 3601–3657.10.2140/gt.2017.21.3601CrossRefGoogle Scholar
Kronheimer, P. B. and Mrowka, T. S., Monopoles and contact structures, Invent. Math. 130 (1997), 209–255.10.1007/s002220050183CrossRefGoogle Scholar
Lipshitz, R., Ozsvath, P. S. and Thurston, D. P., Bordered Heegaard Floer homology, Mem. Amer. Math. Soc. 254 (2018).Google Scholar
Lisca, P. and Stipsicz, A. I., Ozsváth-Szabó invariants and tight contact 3-manifolds. III, J. Symplectic Geom. 5 (2007), 357–384.10.4310/JSG.2007.v5.n4.a1CrossRefGoogle Scholar
Ozsváth, P. and Szabó, Z., Holomorphic disks and genus bounds, Geom. Topol. 8 (2004), 311–334.10.2140/gt.2004.8.311CrossRefGoogle Scholar
Rasmussen, S. D., L-space intervals for graph manifolds and cables, Compos. Math. 153 (2017), 1008–1049.10.1112/S0010437X16008319CrossRefGoogle Scholar
Rasmussen, J. and Rasmussen, S. D., Floer simple manifolds and L-space intervals, Adv. Math. 322 (2017), 738–805.CrossRefGoogle Scholar