On homology cobordism and local equivalence between plumbed manifolds

TL;DR

This paper advances the understanding of involutive Heegaard Floer homology for linear combinations of AR plumbed three-manifolds, establishing new invariance properties and computing correction terms.

Contribution

It proves the Neumann-Siebenmann invariant is a homology cobordism invariant for AR plumbed manifolds and characterizes torsion elements in the cobordism group.

Findings

Neumann-Siebenmann invariant is a homology cobordism invariant for AR plumbed spheres.

Linear combinations with μ(Y)=1 are not torsion in the cobordism group.

Explicit computation of involutive Heegaard Floer correction terms for AR plumbed spaces.

Abstract

We establish a structural understanding of the involutive Heegaard Floer homology for all linear combinations of almost-rational (AR) plumbed three-manifolds. We use this to show that the Neumann-Siebenmann invariant is a homology cobordism invariant for all linear combinations of AR plumbed homology spheres. As a corollary, we prove that if $Y$ is a linear combination of AR plumbed homology spheres with $\mu(Y) = 1$, then $Y$ is not torsion in the homology cobordism group. A general computation of the involutive Heegaard Floer correction terms for these spaces is also included.