Operads, homotopy theory and string topology

运算、同伦理论和弦拓扑

• 批准号: 0405693
• 负责人: James McClure
• 金额: $ 10.8万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405693&HistoricalAwards=false
• 关键词: Operads; homotopy; theory; string; topology

McClure proposes to build on prior work with Jeff Smith which allows severalimportant operads and their actions to be investigated combinatorially(specifically, cosimplicially). One direction, which is already partiallycompleted, is to relate our work to that of Chas and Sullivan. Expectedresults in this direction are: the Chas-Sullivan operations are induced by achain-level action of the framed little 2-disks operad (this is an analogue ofDeligne's Hochschild cohomology conjecture) and the Eilenberg-Moore spectralsequence converging to the homology of the free loop space of a compactoriented PL manifold is a spectral sequence of Batalin-Vilkovisky algebras.Another problem which may be accessible by these methods is Kontsevich's ``higher Deligne conjecture.'' A longer-term goal is to combine Mandell's theorem that E-infinity chain algebras model topological spaces with our combinatorial E-infinity chain operads to obtain explicit results.An interesting development in theoretical physics during the last twenty yearshas been the increasing significance of homotopy theory. It seems to beuseful to create chain-complex models for quantum field theories, both as a preliminary step toward the full development of such theories and as a permanent computational device. Operads have played an important role as anorganizational tool for this kind of work. Recently Chas and Sullivan haveshown that the unbased loop space of any compact oriented PL manifold is aquantum field theory at the level of homology; this raises the question ofwhether their work can be lifted to the chain-complex level and whether it is related to an operad action (specifically, to an action of the framed little disks operad). Techniques developed by McClure and Smith are well-adapted tothis question, and this is one of the directions that McClure intends topursue.

麦克卢尔建议建立在杰夫·史密斯之前工作的基础上，该工作允许对几个重要的歌剧及其行为进行组合研究(具体地说，以协简单的方式)。一个已经部分完成的方向是将我们的工作与查斯和沙利文的工作联系起来。在这一方向的预期结果是：Chas-Sullivan运算是由框架的小2-圆盘算子的链级作用(这是Deligne的Hochschild上同调猜想的类似)诱导的，并且收敛于紧定向PL流形的自由环空间的同调的Eilenberg-Moore谱序列是Batalin-Vilkovisky代数的谱序列。一个较长期的目标是将Mandell关于E-无限链代数建模拓扑空间的定理与我们的组合E-无限链运算符相结合，以获得明确的结果。在过去的二十年中，理论物理中的一个有趣的发展是同伦理论的重要性日益增加。为量子场论创建链状复杂模型似乎是有用的，既是迈向此类理论全面发展的初步步骤，也是一种永久的计算设备。歌剧作为这类工作的组织工具发挥了重要作用。最近，Chas和Sullivan发现，任何紧致定向PL流形的无基环空间都是同调水平上的水量子场论，这就提出了一个问题：他们的工作是否可以提升到链-复数水平，以及它是否与算子作用(具体地说，与框定的小圆盘算子的作用)有关。麦克卢尔和史密斯开发的技术很好地适应了这个问题，这也是麦克卢尔打算探索的方向之一。