Families of contact structures on 3-manifolds; surgery descriptions and open book decompostions of contact manifolds

• Applicant: Professor Dr. Hansjörg Geiges
• Subject Area: Mathematics
• Term: from 2003 to 2008
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 5406692

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.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1154: Global Differential Geometry