Gauge theoretic invariants of 3- and 4-dimensional manifolds

规范 3 维和 4 维流形的理论不变量

• 批准号: 5407083
• 负责人: Professor Dr. Stefan Bauer
• 金额: --
• 依托单位: Fakultät für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2003
• 资助国家: 德国
• 起止时间: 2002-12-31 至 2006-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/5407083?language=en
• 关键词: Gauge; theoretic; invariants; dimensional; manifolds

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.

通过壮观的工作西蒙唐纳森规范理论的方法成为必不可少的考虑时，流形的第四个方面。考虑瞬子模空间的上同调数据，唐纳森定义了同胚流形上区别可微结构的不变量。在类似的精神，塞贝格和维滕定义不变量的基础上模空间的单极。这些塞伯格-威滕不变量被证明是更容易计算，似乎携带相同的信息作为唐纳森的不变量。最近，Bauer和Furuta定义了Seiberg-Witten不变量的稳定同伦加细，它比唐纳森和Seiberg-Witten不变量携带更多的信息。该研究项目的目的是获得这些不同的规范理论不变量的更多结构性理解：不变量在（分歧或非分歧）覆盖下如何表现？他们在家庭中是如何表现的？如何在一个等变的设置？特别是：在剪切和粘贴底层流形时，不变量的行为如何？试图了解剪切和粘贴下的不变量的行为，自然会导致对通常称为Floer同源性的研究。这些群体和与他们相关的三流形不变量是独立的利益，将是该项目的主要重点。