Operads, homotopy theory and string topology

麦克卢尔建议建立在先前的工作与杰夫史密斯允许几个重要的操作和他们的行动进行调查combinatorially（具体地说，cossimplically）。 其中一个方向，已经部分完成，是将我们的工作与查斯和沙利文的工作联系起来。 这方面的预期成果是：Chas-Sullivan运算是由框架小2盘运算的链级作用诱导的（这是Deligne的Hochschild上同调猜想的一个类似）和收敛到紧定向PL流形的自由圈空间的同调的Eilenberg-Moore谱序列是Batalin-Vilkovisky代数的谱序列。高阶Deligne猜想一个长期的目标是将联合收割机Mandell的E-无穷链代数模型拓扑空间的定理与我们的组合E-无穷链操作结合起来，以获得明确的结果。在过去的二十年里，理论物理学中一个有趣的发展是同伦理论的重要性日益增加。 为量子场论建立链复合模型似乎是有用的，既可以作为全面发展这种理论的初步步骤，也可以作为永久的计算设备。 作为一种组织工具，操作在这类工作中发挥了重要作用。 最近Chas和Sullivan已经表明，任何紧致定向PL流形的无基循环空间是在同调水平上的量子场论;这就提出了一个问题，即他们的工作是否可以提升到链复体水平，以及它是否与操作数作用有关（具体地说，与框架小圆盘操作数的作用有关）。 麦克卢尔和史密斯开发的技术很好地适应了这个问题，这是麦克卢尔打算追求的方向之一。

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.