Whitney polygons, symbol homology and cobordism maps

TL;DR

This paper introduces symbol homology, a new framework based on decorated Whitney polygons, to unify and transfer properties across different Heegaard Floer homologies and construct cobordism maps.

Contribution

It develops a novel homology theory using decorated moduli spaces, enabling new presentations and invariance proofs in Heegaard Floer theory.

Findings

Defined symbol homology via decorated Whitney polygons

Established morphisms connecting different Floer homologies

Constructed cobordism maps and proved invariance in knot Floer theories

Abstract

We define a new homology theory we call symbol homology by using decorated moduli spaces of Whitney polygons. By decorating different types of moduli spaces we obtain different flavors of this homology theory together with morphisms between them. Each of these flavors encodes the properties of a different type of Heegaard Floer homology. The morphisms between the symbol homologies enable us to push properties from one Floer theory to a different one. Furthermore, we obtain a new presentation of Heegaard Floer theory in which maps correspond to multiplication from the right with suitable elements of our symbol homology. Finally, we present the construction of cobordism maps in knot Floer theories and apply the tools from symbol homology to give an invariance proof.