Floer Homology and Homology Cobordisms

建议：DMS-9971731PI：Kim Froyhov这个项目涉及定向同调3-球面Y的Yang-Mills和Seiberg-Witten Floer同调群。在他早期关于有理Yang-Mills Floer理论的工作中，主要研究人员已经证明了Y上不可约平坦联络和平凡联络之间的相互作用可以用一个整数来描述。这个整数是Y的同调余边类的一个不变量，在连通和下是可加的。如果Y有界于具有非标准负定交形式的光滑4-流形，则不变量为正。在项目的第一部分，主要研究人员将研究有理Seiberg-Witten Floer理论中相应的不变量，该理论定义为具有自旋c结构的定向有理同调3-球。第二部分将利用PU(2)单极子讨论积分同调球面上这两个不变量之间的关系。本项目第三部分的目的将是理解具有mod 2系数的Yang-Mills Floer理论产生的同调协边不变量。这些不变量在Seiberg-Witten理论中似乎没有任何相似之处。总的来说，首席研究员将使用理论物理中的某些非线性偏微分方程组来研究3维和4维空间。

Proposal: DMS-9971731PI: Kim FroyshovThis project is concerned with Yang-Mills and Seiberg-Witten Floer homology groups of oriented homology 3-spheres Y. In his earlier work on rational Yang-Mills Floer theory the principal investigator has shown that interaction between irreducible flat connections and the trivial connection over Y can be described by a single integer. This integer is an invariant of the homology cobordism class of Y, and is additive under connected sums. The invariant is positive if Y bounds a smooth 4-manifold with non-standard negative definite intersection form. In the first part of the project the principal investigator will study the corresponding invariant in rational Seiberg-Witten Floer theory, which is defined for oriented rational homology 3-spheres with a spin-c structure. The second part will focus on the relationship between these two invariants in the case of integral homology spheres, using PU(2) monopoles. The object of the third part of the project will be to understand the homology cobordism invariants arising from Yang-Mills Floer theory with mod 2 coefficients. These invariants do not seem to have any analogues in Seiberg-Witten theory.Broadly speaking, the principal investigator will study spaces of dimension3 and 4 using certain non-linear partial differential equations from theoretical physics.