Gauge theory, gluing theorems, and their applications

这个项目的目的是证明光滑四流形的规范理论不变量之间的一些猜想关系，并研究是否有可能为b^偶数的四流形定义一族新的不变量。第一步是证明一些关于Taubesian胶合映象的分析结果，该映象模拟了SO(3)单极子模空间的末端。这些结果将完成SO(3)单极共线公式的证明，给出Seiberg-Witten和Donaldson不变量之间的关系，以及Seiberg-Witten和自旋不变量之间的关系。这些关系类似于Witten关于Seiberg-Witten不变量和Donaldson不变量的猜想，不同之处在于它们包含一些仅取决于底层四维流形的拓扑类型的未知系数。第二步推广了Taubes关于反自对偶联络模空间的粘合映射的工作，证明了反自对偶联络模空间的气泡树紧化是一个有边界的流形。这一结果将为证明Donaldson和自旋不变量之间的关系提供一个框架。将这三种关系结合起来，有望对这些规范理论不变量产生新的拓扑约束。最后，对SO(3)单极不变量的分析工作应该回答了这样一个问题：即使使用SO(3)单极的模空间，是否也可以定义具有b^的四维流形的规范理论不变量。流形是局部类似于我们日常经验的欧几里德空间的形状。在(n，d)个变量中的n个方程的解集通常是d维流形：因此，流形出现在整个数学及其应用中，从用于描述DNA的纽结理论到出现在弦理论中的流形。如果两个流形中的一个可以不起皱地伸展到另一个流形中，那么两个流形是异同胚的。确定四维流形何时是微分同胚流形已被证明是一个特别困难的问题。不变量是将诸如数或多项式的代数对象以这样的方式分配给拓扑空间的规则，即当空间拉伸时，代数对象不改变。因此，如果I是不变量，且X和Y是拓扑空间，且I(X)不等于I(Y)，则X不能被拉伸到Y中，因此X和Y不是微分同胚的。这个项目的目标是寻找不同不变量之间的关系，并找到新的不变量。

The goal of this project is to prove some conjectured relations among gauge theoretic invariants of smooth four-manifolds and to investigate whether it is possible to define a new family of invariants for four-manifolds with b^+ even. The first step is to prove some analytic results on the Taubesian gluing maps which model the ends of the moduli space of SO(3) monopoles. These results will complete the proof of the SO(3) monopole cobordism formula, giving a relation between the Seiberg-Witten and the Donaldson invariants and a relation between the Seiberg-Witten and the spin invariants. These relations resemble Witten's conjecture relating the Seiberg-Witten and the Donaldson invariants except that they contain some unknown coefficients depending only on the topological type of the underlying four-manifold. The second step is an extension of Taubes' work on the gluing maps for the moduli space of anti-self-dual connections to prove that the bubbletree compactification of the moduli space of anti-self-dual connections is a manifold with boundary. This result will provide a framework to prove a relation between the Donaldson and spin invariants. It is hoped that combining the three relations will yield new topological constraints on these gauge theoretic invariants. Finally, the analytic work on the SO(3) monopole invariants should answer the question of whether it is possible to define gauge-theoretic invariants for four-manifolds with b^+ even by using the moduli space of SO(3) monopoles.Manifolds are shapes which locally resemble the Euclidean space of our everyday experience. The solution set of n equations in (n+d) variables will usually be a d-dimensional manifold: thus manifolds appear throughout mathematics and its applications, from the knot theory used to describe DNA to the manifolds appearing in string theory. Two manifolds are diffeomorphic if one can be stretched, without wrinkling, into the other. Deciding when four-dimensional manifolds are diffeomorphic has proven to be a particularly difficult problem. An invariant is a rule assigning an algebraic object such as a number or polynomial to a topological space in such a manner that the algebraic object does not change when the space is stretched. Hence if I is an invariant and X and Y are topological spaces with I(X) not equal to I(Y) then X cannot be stretched into Y so X and Y are not diffeomorphic. The goal of this project is to look for relations between different invariants and to find new invariants.