Floer Homology and Homology Cobordisms

建议：DMS-9971731 PI：Kim Froyshov本项目涉及定向同调3-球面Y的Yang-Mills和Seiberg-Witten Floer同调群。在他早期的工作理性杨米尔斯弗洛尔理论的主要研究者表明，相互作用之间的不可约平坦的联系和平凡的联系超过Y可以描述一个单一的整数。这个整数是Y的同调配边类的不变量，并且在连通和下是可加的。如果Y是一个具有非标准负定交形式的光滑4-流形的边界，则该不变量是正的。在该项目的第一部分，主要研究者将研究理性Seiberg-Witten Floer理论中相应的不变量，该理论定义为具有自旋c结构的定向理性同调3-球。第二部分将集中在这两个不变量之间的关系，在积分同调球的情况下，使用PU（2）单极。该项目的第三部分的目的将是了解的同调配边不变量所产生的杨米尔斯弗洛尔理论与模2系数。这些不变量似乎在Seiberg-Witten理论中没有任何类似物。广义地说，主要研究者将使用理论物理中的某些非线性偏微分方程来研究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.