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On homology cobordism and local equivalence between plumbed manifolds

Irving Dai and Matthew Stoffregen

Geometry & Topology 23 (2019) 865–924

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.

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involutive Heegaard Floer homology, homology cobordism
Mathematical Subject Classification 2010
Primary: 57R58
Secondary: 57M27
Received: 10 December 2017
Revised: 22 July 2018
Accepted: 30 August 2018
Published: 8 April 2019
Proposed: András I Stipsicz
Seconded: Ciprian Manolescu, Walter Neumann
Irving Dai
Department of Mathematics
Princeton University
Princeton, NJ
United States
Matthew Stoffregen
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States