## Geometry & Topology

### On homology cobordism and local equivalence between plumbed manifolds

#### 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 $μ ( Y ) = 1$, 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.

#### Article information

Source
Geom. Topol., Volume 23, Number 2 (2019), 865-924.

Dates
Revised: 22 July 2018
Accepted: 30 August 2018
First available in Project Euclid: 17 April 2019

https://projecteuclid.org/euclid.gt/1555466432

Digital Object Identifier
doi:10.2140/gt.2019.23.865

Mathematical Reviews number (MathSciNet)
MR3939054

Zentralblatt MATH identifier
07056055

Subjects
Primary: 57R58: Floer homology
Secondary: 57M27: Invariants of knots and 3-manifolds

#### Citation

Dai, Irving; Stoffregen, Matthew. On homology cobordism and local equivalence between plumbed manifolds. Geom. Topol. 23 (2019), no. 2, 865--924. doi:10.2140/gt.2019.23.865. https://projecteuclid.org/euclid.gt/1555466432

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