Involutive Heegaard Floer homology and plumbed three-manifolds

TL;DR

This paper computes involutive Heegaard Floer homology for plumbed three-manifolds, including Seifert fibered spheres, and explores implications for the homology cobordism group.

Contribution

It provides explicit calculations of involutive Heegaard Floer homology for a broad class of three-manifolds and their connected sums, offering new insights into their algebraic structures.

Findings

Computed involutive Heegaard Floer homology for plumbed manifolds.

Determined involutive correction terms for sums of such manifolds.

Proved the existence of an infinite-rank subgroup in the homology cobordism group.

Abstract

We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involutive correction terms in the case that all of the summands have the same orientation. Using these calculations, we give a new proof of the existence of an infinite-rank subgroup in the three-dimensional homology cobordism group.