Algebraic Structures in Topology

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DMS-0104004Alexander A. VoronovThe goal of the project is to discover and study new algebraicstructures in topology suggested or motivated by mathematical physics, in particular, quantum field theory and string theory. More specifically, the project aims at discovering a new algebraicstructure on the homology of an n-sphere space, by which we mean the space of continuous maps from the n-dimensional sphere to a given manifold. This part of the project, joint with Dennis Sullivan, generalizes the work pioneered by Chas and Sullivan in the case n=1, i.e., that of a usual free loop space. Another goal is to establish connection between Chas-Sullivan's work and Gromov-Witten invariants, which we believe to be a holomorphic version of Chas-Sullivan's algebraic structure. Gromov-Witten invariants come from sigma model of quantum field theory, and Chas-Sullivan's work "String Topology" may be regarded as a topological version of the physical construction. This part of the project is suggested to be completed by developing a fusion intersection theory of semi-infinite cycles in infinite dimensional manifolds. Finally, part of the project isdedicated to relating the above to Kontsevich's Conjecture, which generalizes Deligne's Conjecture and unravels a deep relation between deformation theory of abstract n-algebras and the topology of configuration spaces of points in an (n+1)-dimensional Euclidean space.The main idea of Algebraic Topology is to be able to recognizetopological properties of a geometric object by associating algebraic data or structure to the geometric object. Sometimes the geometry is too complicated to allow immediate understanding and work with the object, while the algebraic information is usually simpler by its nature. This project suggests some new algebraic structure for a sphere space, the space of maps from an n-dimensional sphere to a manifold. Such spaces are quite complicated and the classical work of Chen, Segal, Jones, Getzler, Burghelea, Fedorowicz, Goodwillie, and others, produced not only the computation of the homology of loop spaces, which are the particular case of sphere spaces for n=1, but also revealed amazing connections with algebra (Hochschildcomplex). Also, recent progress in string theory emphasized the importance of invariants associated to holomorphic maps from the 2-sphere to a manifold (Gromov-Witten invariants). In this project we undertake an analogous study of continuous maps from the n-sphere to a manifold, which for n=1 has already enabled significant progress in topology.