Naturality and functoriality in involutive Heegaard Floer homology

TL;DR

This paper establishes the naturality and functoriality of involutive Heegaard Floer homology, enabling well-defined cobordism maps and extending these properties to knots and links, advancing the theoretical framework of Floer homology.

Contribution

It proves naturality and functoriality for involutive Heegaard Floer homology and constructs associated cobordism maps, extending these results to knots and links.

Findings

Proved first-order naturality of involutive Heegaard Floer homology.

Constructed well-defined cobordism maps for involutive Floer theory.

Extended naturality and functoriality results to knots and links.

Abstract

We prove first-order naturality of involutive Heegaard Floer homology, and furthermore construct well-defined maps on involutive Heegaard Floer homology associated to cobordisms between three-manifolds. We also prove analogous naturality and functoriality results for involutive Floer theory for knots and links. The proof relies on the doubling model for the involution, as well as several variations.