Higher-dimensional Heegaard Floer homology and Hecke algebras

Abstract

Given a closed oriented surface Σ of genus greater than 0, we construct a map F from the higher-dimensional Heegaard Floer homology of the cotangent fibers of T * Σ to the Hecke algebra associated to Σ and show that F is an isomorphism of algebras. We also establish analogous results for punctured surfaces. Contents 1. Introduction 1 2. Wrapped higher-dimensional Heegaard Floer cohomology 6 2.1. Review of HDHF 6 2.2. Wrapped HDHF of disjoint cotangent fibers 8 3. The relationship between the loop space and the wrapped Floer homology of a cotangent fiber 11 4. Hecke algebras 14 5. The parameter c in HDHF 17 6. The evaluation map 19 6.1. The definition 19 6.2. The isomorphism 22 7. Surfaces with punctures 28 References 32