Foliations and contact structures
Final Report Abstract
The primary goal of this research project was the study of the relationship between contact structures and foliations an closed 3-manifolds. We recapitulate the main facts which were previously known and compare them with results we found. (1) Y. Eliashberg and W.Thurston proved that every C2-foliation without spherical leaves can be C°-approximated by contact structures. While the assumption an spherical leaves cannot be removed it is a drawback, that only C2-foliations are covered since many interesting constructions of foliations yield foliations with lower regularity. J. Bowden[B]improved this result so that it holds for C°-foliations. This was also achieved independently by R. Roberts and W. Kazez. (2) V. Colin and S. Firmo study pairs of contact structures(one of them positive, the other negative) which are transverse (or paus of contact satisfying a weaker, but more technical condition). They show that from a pair (ζ+,ζ_) an M3 one can obtain a new plane field which is nowhere a contact structure and which gives rise to a foliation under additional assumptions. One of their main results is that if both contact structure are tight, then the resulting foliation has no Reeb component whose soul represents atrivial class in H1 (M;Q). We can now also show that if both contact structures are universally and the plane field obtained from this pair yields a foliation, then this foliation has no Reeb component at all. This can be considered as a first step to giving a criterion for the existence of a taut foliation an a closed 3-manifold purely in terms of contact topology. (3) J. Bowden started considering Anosov flows and flows satisfying weaker hyperbolicity assumptions and how they relate. In work in progress, he attempts relate properties of foliations and contact paus associated to a flow such as tightness and tautness to that of the existence of an Anosov flow. (4) Beyond this the work concerning extensions of Eliashberg and Thurston's result to foliations of lower regularity motivates questions about the existence of taut foliations with additional smoothness assumptions.. This question is then resolved in the case of graph manifolds in joint work with S. Boyer. (5) Furthermore, ideas about relating dynamics and contact structures using certain flows lead to new proof of certain stability phenomenon for certain kinds of actions an the circle. These proofs haue the advantage of extending easily to higher dimensions which is the content of Joint work with K. Mann. (6) Although its results do not relate directly to the research proposal, we would like to mention Vogel's paper because it continuous to develop methods which might be relevant to the research project.
Publications
- Approximating C°-foliations by contact structures, Geom. Funct. Anal., 2 (2016) 1255-1296
J. Bowden
(See online at https://doi.org/10.1007/s00039-016-0387-2) - On the uniqueness of the contact structure approximating a foliation, Geom. Topol. 20(2016), no. 5, 2439-2573
T. Vogel
(See online at https://dx.doi.org/10.2140/gt.2016.20.2439) - C° approximations of foliations, Geom. Topol. 21 (2017), no. 6, 3601-3657
W. Kazez, R. Roberts
(See online at https://dx.doi.org/10.2140/gt.2017.21.3601) - Non-loose unknots, overtwisted discs and the contact mapping class group of S3, Geom. Funct. Anal. 28 (2018), no. 1, 228-288
T. Vogel
(See online at https://doi.org/10.1007/s00039-018-0439-x)
