Gauge theoretic invariants of 3- and 4-dimensional manifolds

• Applicant: Professor Dr. Stefan Bauer
• Subject Area: Mathematics
• Term: from 2003 to 2007
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 5407083

Through the spectacular work of Simon Donaldson gauge theoretic methods became indispensable when considering manifolds in dimension four. Considering cohomological data of moduli spaces of instantons, Donaldson defined invariants which could distinguish differentiable structures on homeomorphic manifolds. In a similar spirit, Seiberg and Witten defined invariants based on the moduli spaces of monopoles. These Seiberg-Witten invariants turned out to be far easier to compute, seemingly carrying the same information as Donaldson's invariants. Recently, Bauer and Furuta defined a stable cohomotopy refinement of Seiberg-Witten invariants, which turns out to carry more information than both Donaldson and Seiberg-Witten invariants. The aim of the research project is to gain a more structural understanding of these different gauge theoretic invariants: How do the invariants behave under (ramified or unramified) coverings? How do they behave in families? How in an equivariant setting? In particular: How do the invariants behave under cutting and pasting of the underlying manifolds? Attempting to understand the behaviour of the invariants under cutting and pasting leads naturally to a study of what is commonly known as Floer homology. These groups and three-manifold invariants related to them are of independent interest and will be a main focus of the project.

• DFG Programme: Research Grants
• Participating Person: Dr. Kim Froyshov