The fully marked surface theorem
In his seminal 1976 paper Bill Thurston observed that a closed leaf S of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. The main result of this paper is a converse for taut foliations: if the Euler cl...
| Main Authors: | , |
|---|---|
| Format: | Working paper |
| Language: | English |
| Published: |
University of Oxford
2020
|
| _version_ | 1797102276832133120 |
|---|---|
| author | Gabai, D Yazdi, M |
| author_facet | Gabai, D Yazdi, M |
| author_sort | Gabai, D |
| collection | OXFORD |
| description | In his seminal 1976 paper Bill Thurston observed that a closed leaf S of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation F evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation F′ such that S is homologous to a union of leaves and such that the plane field of F′ is homotopic to that of F. In particular, F and F′ have the same Euler class.
In the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. This is the second of two papers that together give a negative answer to Thurston's conjecture. In the first paper, counterexamples were constructed assuming the main result of this paper. |
| first_indexed | 2024-03-07T06:03:40Z |
| format | Working paper |
| id | oxford-uuid:ed177d06-9111-4b03-8e2b-7f152ae2a47f |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T06:03:40Z |
| publishDate | 2020 |
| publisher | University of Oxford |
| record_format | dspace |
| spelling | oxford-uuid:ed177d06-9111-4b03-8e2b-7f152ae2a47f2022-03-27T11:22:22ZThe fully marked surface theoremWorking paperhttp://purl.org/coar/resource_type/c_8042uuid:ed177d06-9111-4b03-8e2b-7f152ae2a47fEnglishSymplectic ElementsUniversity of Oxford2020Gabai, DYazdi, MIn his seminal 1976 paper Bill Thurston observed that a closed leaf S of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation F evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation F′ such that S is homologous to a union of leaves and such that the plane field of F′ is homotopic to that of F. In particular, F and F′ have the same Euler class. In the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. This is the second of two papers that together give a negative answer to Thurston's conjecture. In the first paper, counterexamples were constructed assuming the main result of this paper. |
| spellingShingle | Gabai, D Yazdi, M The fully marked surface theorem |
| title | The fully marked surface theorem |
| title_full | The fully marked surface theorem |
| title_fullStr | The fully marked surface theorem |
| title_full_unstemmed | The fully marked surface theorem |
| title_short | The fully marked surface theorem |
| title_sort | fully marked surface theorem |
| work_keys_str_mv | AT gabaid thefullymarkedsurfacetheorem AT yazdim thefullymarkedsurfacetheorem AT gabaid fullymarkedsurfacetheorem AT yazdim fullymarkedsurfacetheorem |