Applications of involutive Heegaard Floer homology

TL;DR

This paper introduces a new invariant derived from involutive Heegaard Floer homology to analyze the homology cobordism group, providing computational tools and insights into the structure of three-manifolds.

Contribution

It defines a novel homology cobordism invariant using involutive Heegaard Floer homology, extending previous invariants and enabling new structural analyses.

Findings

Computed involutive correction terms for specific three-manifold families

Provided new insights into the structure of the homology cobordism group

Established connections with Stoffregen's connected Seiberg-Witten Floer homology

Abstract

We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to Stoffregen's connected Seiberg-Witten Floer homology. We use this invariant to study the structure of the homology cobordism group and, along the way, compute the involutive correction terms for certain families of three-manifolds.