Operads, homotopy theory and string topology

点击翻译按钮获取中文摘要

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.