Transversal String Topology and Invariants of Manifolds

Abstract

Loop spaces have played a recurring and important role in mathematics - from closed geodesics in differential geometry to based loop spaces which play a central role in homotopy theory. The subject of string topology originating from the seminal paper by Chas and Sullivan, focuses on topological aspects of the free loop space of manifolds; it's the study of the algebraic structures present therein. From the point of view of computations, several techniques of algebraic topology may apply. We show, using rational homotopy theory and minimal models, that the Lie algebra structure on the (circle) equivariant homology of a product of odd spheres is highly non-trivial although the same structure for an odd sphere is trivial. Similar smaller computational results are also presented. In the main result of this work, we define and study certain geometric loops, called transversal strings, which satisfy some specific boundary conditions. The relevant algebraic backdrop happens to be the category of bicomodules and algebra objects in this setting. Using the machinery of minimal models and homological algebra in this setting, we show that using transversal string topology it's possible to distinguish non-homeomorphic but homotopy equivalent Lens spaces.