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:
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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.
-
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).
-
F. Ding, H. Geiges and A.I. Stipsicz, Lutz twist and contact surgery,
Asian J. Math. 9 (2005), 57-64.
-
F. Ding and H. Geiges, E8-plumbings and exotic contact
structures on spheres,
Int. Math. Res. Not. (2004), no. 71, 3825-3837.
-
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.
-
F. Ding and H. Geiges, Legendrian knots and links classified by
classical invariants,
Commun. Contemp. Math. 9 (2007), 135-162.
-
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.
-
H. Geiges, Contact Dehn surgery, symplectic fillings, and property P for
knots,
Expo. Math. 24 (2006), 273-280.
-
H. Geiges and F. Pasquotto, A formula for the Chern classes of symplectic
blow-ups,
J. London Math. Soc. (2) 76 (2007), 313-330.
-
M. Zessin,
On contact tops and integrable tops,
Indag. Math. (N.S.) 18 (2007), 305-325.
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F. Ding and H. Geiges, A unique decomposition theorem for tight contact
3-manifolds,
Enseign. Math. (2) 53 (2007), 333-345.
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Horizontal loops in Engel space,
Math. Ann. 342 (2008), 291-296.
-
A contact geometric proof of the Whitney-Graustein theorem,
Enseign. Math. (2) 55 (2009), 93-102.
-
(with J. Gonzalo) Contact spheres and hyperkähler geometry,
Comm. Math. Phys. 287 (2009), 719-748.
-
(with F. Ding) Handle moves in contact surgery diagrams,
J. Topol. 2 (2009), 105-122.
-
(with J. Gonzalo) A homogeneous Gibbons-Hawking ansatz and
Blaschke products,
Adv. Math. 225 (2010), 2598-2615.