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Existence of Taut Foliations on Seifert Fibered Homology 3-spheres

Published online by Cambridge University Press:  20 November 2018

Shanti Caillat-Gibert  and
Daniel Matignon
Shanti Caillat-Gibert
Affiliation:
Centre de Mathématiques et Informatique, Aix-Marseille Université, 13453 Marseille Cedex 13, France e-mail: [email protected] [email protected]
Daniel Matignon
Affiliation:
Centre de Mathématiques et Informatique, Aix-Marseille Université, 13453 Marseille Cedex 13, France e-mail: [email protected] [email protected]
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Abstract

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This paper concerns the problem of existence of taut foliations among 3-manifolds. From the work of David Gabai we know that a closed 3-manifold with non-trivial second homology group admits a taut foliation. The essential part of this paper focuses on Seifert fibered homology 3-spheres. The result is quite different if they are integral or rational but non-integral homology 3-spheres. Concerning integral homology 3-spheres, we can see that all but the 3-sphere and the Poincaré 3-sphere admit a taut foliation. Concerning non-integral homology 3-spheres, we prove there are infinitely many that admit a taut foliation, and infinitely many without a taut foliation. Moreover, we show that the geometries do not determine the existence of taut foliations on non-integral Seifert fibered homology 3-spheres.

Keywords

57M2557M5057N1057M15homology 3-spherestaut foliationSeifert-fibered 3-manifolds
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 1 , February 2014 , pp. 141 - 169
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Brittenham, M., Essential laminations in Seifert-fibered spaces. Topology 32(1993), no. 1, 61–85.http://dx.doi.org/10.1016/0040-9383(93)90038-W Google Scholar
[2] Brittenham, M., Naimi, R., and Roberts, R., Graph manifolds and taut foliations. J. Differential Geom. 45(1997), no. 3, 446–470.Google Scholar
[3] Caillat-Gibert, S. and Matignon, D., Compact leaves in Reebless or taut foliations. arxiv:1105.3103v2 Google Scholar
[4] Calegari, D., Promoting essential laminations. Invent. Math. 166(2006), no. 3, 583–643. http://dx.doi.org/10.1007/s00222-006-0004-3 Google Scholar
[5] Clauss, E., Essential laminations in closed Seifert-fibered spaces. Thesis, University of Texas at Austin, 1991.Google Scholar
[6] Eisenbud, D., Hirsch, U., and Neumann, W., Transverse foliations on Seifert bundles and self-homeomorphisms of the circle. Comment. Math. Helvetici 56(1981), no. 4, 638–660. http://dx.doi.org/10.1007/BF02566232 Google Scholar
[7] Eliashberg, Y. and Thurston, W. P., Confoliations. University Lecture Series, 13, American Mathematical Society, Providence, RI, 1998.Google Scholar
[8] Epstein, D. B. A., Periodic flows on three-manifolds. Ann. of Math. 95(1972), 66–82. http://dx.doi.org/10.2307/1970854 Google Scholar
[9] Gabai, D., Foliations and the topology of 3-manifolds. J. Differential Geometry. 18(1983), no. 3, 445–503.Google Scholar
[10] Jankins, M. and Neumann, W., Rotation numbers of products of circle homeomorphisms. Math. Ann. 271(1985), no. 3, 381–400. http://dx.doi.org/10.1007/BF01456075 Google Scholar
[11] Levitt, G., Feuilletages des variétés de dimension 3 qui sont des fibres en cercles. Comment. Math. Helv. 53(1978), no. 4, 572–594. http://dx.doi.org/10.1007/BF02566099 Google Scholar
[12] Lisca, P. and Matić, G., Transverse contact structures on Seifert 3-manifolds. Algebr. Geom. Topol. 4(2004), 1125–1144. http://dx.doi.org/10.2140/agt.2004.4.1125 Google Scholar
[13] Lisca, P. and Stipsicz, A. I., Ozsváth-Szabó invariants and tight contact 3-manifolds, III. J. Symplectic Geom. 5(2007), no. 4, 357–384.Google Scholar
[14] Matsumoto, S., Foliations of Seifert-fibered spaces over S2. In: Foliations (Tokyo, 1983), Adv. Stud. Pure Math., 5, North-Holland, Amsterdam, 1985, pp. 325–339.Google Scholar
[15] Naimi, R., Foliations transverse to fibers of Seifert manifolds. Comment. Math. Helv. 69(1994), no. 1, 155–162. http://dx.doi.org/10.1007/BF02564479 Google Scholar
[16] Novikov, S. P., Foliations of codimension 1 on manifolds. (Russian) Dokl. Akad. Nauk SSSR 155(1964), 1010–1013.Google Scholar
[17] Palmeira, C., Open manifolds foliated by planes. Ann. Math. 107(1978), no. 1, 109–131. http://dx.doi.org/10.2307/1971256 Google Scholar
[18] Rustamov, R., On Heegard Floer homology of three-manifolds. Ph.D. Dissertation, Princeton University, 2005. arxiv:math/0505349 Google Scholar
[19] Roberts, R., Shareshian, J., and Stein, M., Infinitely many hyperboloic 3-manifolds which contain no Reebless foliation. J. Amer. Math. Soc. 16(2003), no. 3, 639–679. http://dx.doi.org/10.1090/S0894-0347-03-00426-0 Google Scholar
[20] Saveliev, N., Invariants for homology 3-spheres. In: Encyclopaedia of Mathematical Sciences, 140, Low-Dimensional Topology, I, Springer-Verlag, Berlin, 2002.Google Scholar
[21] Scott, P., The geometry of 3-manifolds. Bull. London Math. Soc. 15(1983), no. 5, 401–487. http://dx.doi.org/10.1112/blms/15.5.401 Google Scholar
[22] Seifert, H., Topologie dreidimensionaler gefaserter Raume. Acta Math. 60(1933), no. 1, 147–238. http://dx.doi.org/10.1007/BF02398271 Google Scholar
[23] Seifert, H. and Threlfall, W., Topology of 3-dimensional fibered spaces. Pure and Applied Mathematics, 89, Academic Press, 1980, pp. 360–418.Google Scholar
[24] Thurston, W. P., Foliations of 3-manifolds which are circle bundles. Thesis, University of California at Berkeley, 1972.Google Scholar
[25] Waldhausen, F., Eine classe von 3-dimensionalen mannigfaltigkeiten I. Invent. Math. 3(1967), 308–333.Google Scholar