Composition Maps in Heegaard Floer Homology

TL;DR

This paper proves that the opposite composition map in Heegaard Floer homology aligns with the pair-of-pants cobordism map up to homotopy, and provides an algorithm for computing cobordism maps.

Contribution

It establishes the equivalence of composition and cobordism maps in Heegaard Floer homology and introduces a computational algorithm for cobordism maps.

Findings

Opposite composition map agrees with pair-of-pants cobordism map up to homotopy.

Provides an explicit algorithm for computing cobordism maps.

Enhances computational tools in Heegaard Floer homology.

Abstract

We use results of Auroux arXiv:1001.4323 and Zemke arXiv:1801.09270 to prove that, in the morphism spaces formulation of Heegaard Floer homology given in arXiv:1005.1248, the opposite composition map agrees up to homotopy with the map on Heegaard Floer complexes induced by a pair-of-pants cobordism. As an application, we give an algorithm for computing arbitrary cobordism maps on hat Heegaard Floer homology.