Canonical orientations in Heegaard Floer theory

TL;DR

This paper develops Heegaard Floer theory over the integers by establishing canonical orientations from coupled Spin structures, proving naturality, and providing new proofs and definitions within the framework.

Contribution

It introduces a new approach to Heegaard Floer theory over integers using canonical orientations, and proves naturality and related results in this setting.

Findings

Heegaard Floer homology is natural over $\\mathbb{Z}$.

Provides a new proof of the surgery exact triangle.

Defines involutive Heegaard Floer homology over $\\mathbb{Z}$.

Abstract

We set up Heegaard Floer theory over the integers, using canonical orientations coming from coupled Spin structures on the Lagrangian tori. We prove naturality of Heegaard Floer homology, sutured Floer homology, and link Floer homology over $\mathbb{Z}$. We give a new proof of the surgery exact triangle in this context, as well as a definition of involutive Heegaard Floer homology over $\mathbb{Z}$.