Operads, homotopy theory and string topology

McClure建议以Jeff Smith之前的工作为基础，允许对几个重要的操作符及其动作进行组合研究（具体来说，是协简化）。一个已经部分完成的方向是将我们的工作与查斯和苏利文的工作联系起来。在这个方向上的预期结果是：Chas-Sullivan操作是由框架小2盘操作的链级作用诱导的（这是deligne的Hochschild上同调猜想的类似物），并且收敛到紧取向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。另一个可以用这些方法解决的问题是康采维奇的“高德列涅猜想”。“我们的长期目标是将曼德尔关于e -无穷链代数模型拓扑空间的定理与我们的组合e -无穷链操作数结合起来，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。为量子场理论创建链复杂模型似乎是有用的，既可以作为这些理论全面发展的初步步骤，也可以作为永久的计算设备。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证明了任意紧致PL流形的非基环空间在同调水平上是量子场论；这就提出了一个问题，即他们的工作是否可以提升到链复杂级别，以及它是否与操作操作（具体来说，与框架小磁盘操作操作）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.