Contact circles and surgery


Antragsteller: Hansjörg Geiges

Finanzierung: Deutsche Forschungsgemeinschaft (DFG)

Programm: Schwerpunktprogramm Globale Differentialgeometrie

Laufzeit: 6/03-3/08

Förderung der Zusammenarbeit mit: Fan Ding (Beijing), Jesús Gonzalo (Madrid), András Stipsicz (Budapest), Charles Thomas (Cambridge); post-doc Stelle (Dr. Mathias Zessin, 9/05-8/07)

Zusammenfassung: This project studies certain families of contact structures, so-called contact circles and contact spheres, on 3-dimensional manifolds. The aim is to understand the relation of these structures to the Teichmüller theory of complex structures on surfaces, the dynamics of special flows on 3-manifolds, and constructions of hyperkähler metrics arising in physics such as the Gibbons-Hawking ansatz. The ultimate goal is to develop contact circles as a tool for answering questions arising in those areas. Specific aims are to classifiy and understand the geometry of transversely conformal flows on 3-manifolds, to study a generalisation of the Gauß-Bonnet theorem arising from contact circles, and to investigate a generalisation of spin structures to higher orders and orbifolds, and related coverings of Teichmüller space.

The part of the project concerned with contact surgery on 3-manifolds aims to find explicit surgery presentations for contact 3-manifolds and applications of these presentations to questions arising in contact topology.

A third strand of the project is concerned with the existence and classification of contact structures on higher-dimensional manifolds. Particular stress is laid on the existence on spheres of such structures that are compatible with finite group actions on that sphere.


Publikationen:

  1. H. Geiges and J. Gonzalo, On the topology of the space of contact structures on torus bundles,
    Bull. London Math. Soc. 36 (2004), 640-646.

  2. F. Ding, H. Geiges and A.I. Stipsicz, Surgery diagrams for contact 3-manifolds,
    Turkish J. Math. 28 (2004), 41-74 (Proceedings of the 10th Gökova Geometry-Topology conference).

  3. F. Ding, H. Geiges and A.I. Stipsicz, Lutz twist and contact surgery,
    Asian J. Math. 9 (2005), 57-64.

  4. F. Ding and H. Geiges, E8-plumbings and exotic contact structures on spheres,
    Int. Math. Res. Not. (2004), no. 71, 3825-3837.

  5. F. Ding and H. Geiges, Examples of Legendrian knots and links classified by classical invariants,
    in: Proceedings of the Second East Asian School of Knots and Related Topics in Geometric Topology
    (Dalian, China, 2005), 21-24.

  6. F. Ding and H. Geiges, Legendrian knots and links classified by classical invariants,
    Commun. Contemp. Math. 9 (2007), 135-162.

  7. H. Geiges, Contact Dehn surgery, symplectic fillings, and property P for knots,
    in: Arbeitstagung 2005, Max-Planck-Institut für Mathematik, Bonn,
    MPIM Preprint Series 60 (2005), 7 pages,
    ArXiv math.SG/0506472.

  8. H. Geiges, Contact Dehn surgery, symplectic fillings, and property P for knots,
    Expo. Math. 24 (2006), 273-280.

  9. H. Geiges and F. Pasquotto, A formula for the Chern classes of symplectic blow-ups,
    J. London Math. Soc. (2) 76 (2007), 313-330.

  10. M. Zessin, On contact tops and integrable tops,
    Indag. Math. (N.S.) 18 (2007), 305-325.

  11. F. Ding and H. Geiges, A unique decomposition theorem for tight contact 3-manifolds,
    Enseign. Math. (2) 53 (2007), 333-345.

  12. Horizontal loops in Engel space,
    Math. Ann. 342 (2008), 291-296.

  13. A contact geometric proof of the Whitney-Graustein theorem,
    Enseign. Math. (2) 55 (2009), 93-102.

  14. (with J. Gonzalo) Contact spheres and hyperkähler geometry,
    Comm. Math. Phys. 287 (2009), 719-748.

  15. (with F. Ding) Handle moves in contact surgery diagrams,
    J. Topol. 2 (2009), 105-122.

  16. (with J. Gonzalo) A homogeneous Gibbons-Hawking ansatz and Blaschke products,
    Adv. Math. 225 (2010), 2598-2615.