Gauge theoretic invariants of 3- and 4-dimensional manifolds

通过西蒙·唐纳森的杰出工作，规范理论方法在考虑四维流形时变得不可或缺。考虑实例模空间的上同调数据，Donaldson定义了能区分同胚流形上可微结构的不变量。本着类似的精神，Seiberg和Witten基于单极子的模空间定义了不变量。这些Seiberg-Witten不变量更容易计算，似乎与Donaldson不变量携带相同的信息。最近，Bauer和Furuta定义了Seiberg-Witten不变量的稳定同伦细化，它比Donaldson和Seiberg-Witten不变量携带更多的信息。该研究项目的目的是获得对这些不同的规范理论不变量的更结构化的理解：不变量在（分支化或非分支化）覆盖下的行为如何？他们在家庭中表现如何？在等变情况下呢？特别是：不变量在剪切和粘贴基础流形时的行为如何？试图理解剪切和粘贴过程中不变量的行为，自然会导致对通常被称为花同源性的研究。这些组和与它们相关的三流形不变量具有独立的兴趣，并且将成为项目的主要焦点。

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.