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# Existence of Taut Foliations on Seifert Fibered Homology $3$-spheres

Published:2013-11-08
Printed: Feb 2014
• Shanti Caillat-Gibert,
CMI, Aix-Marseille Université
• Daniel Matignon,
CMI, Aix-Marseille Université
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## Abstract

This paper concerns the problem of existence of taut foliations among $3$-manifolds. Since the contribution of David Gabai, we know that closed $3$-manifolds with non-trivial second homology group admit 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 which admit a taut foliation, and infinitely many without 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: homology 3-spheres, taut foliation, Seifert-fibered 3-manifolds
 MSC Classifications: 57M25 - Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M50 - Geometric structures on low-dimensional manifolds 57N10 - Topology of general $3$-manifolds [See also 57Mxx] 57M15 - Relations with graph theory [See also 05Cxx]

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