String Topology and the Algebraic Topology of Moduli Spaces

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This proposal consists of several projects using algebraic topological techniques to study geometric questions arising in String Topology, and the topology of moduli spaces. The theory of String Topology, first introduced by Chas and Sullivan in 1999, now involves a vast array of rich structure on spaces of paths and loops of manifolds, as well as maps from surfaces to manifolds. In this proposal, Cohen will emphasize applications of string topology, and the recent breakthroughs in the study of the homotopy type of moduli spaces. These applications will be both to geometry (understanding the Gromov-Witten theory of cotangent bundles, and the cobordism type of moduli spaces of holomorphic curves), and within algebraic topology (the K -theory of loop spaces of classifying spaces, twisted equivariant K -theory, and Waldhausen's algebraic K -theory of spaces). Cohen, in collaboration with I. Madsen, will also study the homotopy type of moduli spaces of maps of surfaces, extending Madsen and Weiss's recent work on the generalized Mumford conjecture. They propose a longer term project to use this knowledge, as well as an adaptation of string topology methods, to understand bordism classes represented by the compact moduli spaces of stable holomorphic curves in a symplectic manifold. This proposal consists of several projects investigating the new area of research known as "String Topology", as well as related questions. String topology, a theory that was first introduced by Chas and Sullivan in 1999, studies structures on spaces of paths, loops, and surfaces. This structure was motivated by formalisms in string theory in physics. The idea is to understand how loops (or paths) in a background space can evolve in time. Loops can evolve by changing in size and even breaking apart. These phenomena are measured by studying surfaces mapping to the background space, that span these loops. In this project, Cohen tends to study the intersection properties of these spaces of loops and surfaces, and in particular the topology of the space of surfaces mapping to a background space. His goal is to apply these constructions to a variety of geometric questions.