Mathematical Sciences: Construction of a Geometric Category Representing H4(M;Z), and Its Implications

在接下来的三年里，McLaughlin计划找到一个表示H4(M；Z)的范畴，其方式类似于将H2(M；Z)解释为线丛的同构类，将H3(M；Z)解释为Gerb的等价类。他将研究弦理论、椭圆上同调、代数K-理论、纽结理论的结果，以及定义三次非交换上同调的问题。其基本思想是使用一个光滑版本的Deligne上同调，并将其越位到自由环空间。这将被解释为广义完整映射。该范畴将通过推导出哪种几何结构具有给定的GERB作为其完整概念来构建。麦克劳夫林用这种方式发现了代表庞特贾金阶级的物体。这项工作将为Deligne上同调提供一个新的几何视角，因为所构造的对象实际上位于这个更精细的理论中。反过来，使用贝林森调节器的代数K理论将会产生后果。越界的存在增加了陈氏迭代积分与Deligne上同调之间的深层次联系的可能性，提炼了Getzler，Jones和Petrick的工作。麦克劳克林已经得到了这一方向的初步公式，这些公式强调了杯子产品所发挥的非凡作用。该程序为椭圆上同调中的几何上循环提供了一个候选者。这个物体是一个与GERB相关联的“假的”无限维矢量束。它在当地看起来像一个媒介束，但以不同的方式粘合在一起。这种方法的优点是能够协调所有关于椭圆上同调的标准猜想，并将是拟议研究中的优先事项。这个“伪”丛也可以作为循环空间上的字符串丛的替代品来研究，从而为严格的字符串索引理论开辟了道路。最后，在三维流形M的情况下，这个对象实际上超越了在M中的纽结空间上给出的新结构。我们将充分研究纽结理论的含义。该项目的总体目标是统一几个深层次且目前完全不同的拓扑学领域。这将通过展示他们研究中使用的分析技术是共同祖先的不同方面来完成的。展望了纽结理论以及几何学和拓扑学的其他活跃领域的应用。

Over the next three years, McLaughlin plans to find a category representing H4(M;Z), in a manner similar to the interpretation of H2(M;Z) as isomorphism classes of line bundles, and H3(M;Z) as equivalence classes of gerbs. He will investigate the consequences for string theory, elliptic cohomology, algebraic K-theory, knot theory, and the problem of defining degree three, non-abelian, sheaf cohomology. The basic idea is to use a smooth version of Deligne cohomology, and the transgression to the free loop space. This will be interpreted as a generalized holonomy map. The category will be constructed by deducing which geometric structure has a given gerb as its holonomy. McLaughlin discovered the object representing the first Pontrjagin class in this way. This work will give a new geometric insight into Deligne cohomology, as the objects constructed actually lie in this more refined theory. There will in turn, be consequences for algebraic K-theory, using Beilinson's regulator. The very existence of the transgression raises the possibility of a deep relationship between Chen's iterated integrals and Deligne cohomology, refining the work of Getzler, Jones and Petrack. McLaughlin has obtained preliminary formulae in this direction, which emphasize the extraordinary role played by the cup product. This program offers a candidate for the geometric cocycle in elliptic cohomology. The object in question is a "fake" infinite-dimensional vector bundle associated to a gerb. It looks locally like a vector bundle but glues together in a different way. This approach has the advantage of reconciling all the standard conjectures about elliptic cohomology and will be a priority in the proposed research. This "fake" bundle can also be studied as a substitute for the string bundle on loop space, opening the way for a rigorous index theory of strings. Finally, in the case of a three- manifold M, this object actually transgresses to give a new structure on the space of knots in M. The implications for knot theory will be fully investigated. The overall objective of this project is to unify several deep and currently disparate areas of topology. This will be done by showing how analytic techniques employed in their study are different aspects of a common ancestor. Applications are envisioned to the theory of knots and to other active areas of geometry and topology.