Hecke Algebras Via Morse Theory of Loop Spaces

Higher-dimensional Heegaard Floer homology (HDHF) is defined by extending Lipshitz's cylindrical reformulation of Heegaard Floer homology from surfaces to arbitrary Liouville domains. The HDHF also serves as a model for Lagrangian Floer homology of symmetric products.

In this talk, I will present a Morse-theoretic model allowing for computations of the HDHF A∞-algebra of k cotangent fibers in the cotangent bundle of a smooth manifold. We apply this model to get an explicit computation of this A∞-algebra for the cotangent bundle of the 2-dimensional sphere. The result of this computation produces a differential graded algebra which may be regarded as the derived HOMFLYPT skein algebra of the sphere.