Existence of Taut foliations on Seifert fibered homology 3-spheres

TL;DR

This paper investigates the existence of taut foliations on Seifert fibered homology 3-spheres, revealing that their existence depends on whether the spheres are integral or non-integral, with many admitting or not admitting such foliations.

Contribution

It provides a comprehensive classification of taut foliation existence on Seifert fibered homology 3-spheres, distinguishing between integral and non-integral cases and showing the influence of geometry.

Findings

All but the 3-sphere and Poincaré sphere admit taut foliations among integral homology 3-spheres.

Infinitely many non-integral homology 3-spheres admit taut foliations, and infinitely many do not.

The geometry of the sphere does not determine the existence of taut foliations in non-integral cases.

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 foliations. 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 prove that all but the 3-sphere and the Poincar\'e 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.