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Lattice cohomology and q-series invariants of 3-manifolds

  • Rostislav Akhmechet ORCID logo , Peter K. Johnson ORCID logo und Vyacheslav Krushkal ORCID logo EMAIL logo
Veröffentlicht/Copyright: 23. Februar 2023

Abstract

In this paper, an invariant is introduced for negative definite plumbed 3-manifolds equipped with a spin c -structure. It unifies and extends two theories with rather different origins and structures. One theory is lattice cohomology, motivated by the study of normal surface singularities, known to be isomorphic to the Heegaard Floer homology for certain classes of plumbed 3-manifolds. Another specialization gives BPS q-series which satisfy some remarkable modularity properties and recover SU ( 2 ) quantum invariants of 3-manifolds at roots of unity. In particular, our work gives rise to a 2-variable refinement of the Z ^ -invariant.

Award Identifier / Grant number: DMS-1839968

Award Identifier / Grant number: DMS-2105467

Funding source: Simons Foundation

Award Identifier / Grant number: 608604

Funding statement: Rostislav Akhmechet was supported by NSF RTG grant DMS-1839968, NSF grant DMS-2105467 and the Jefferson Scholars Foundation. Peter K. Johnson was supported by NSF RTG grant DMS-1839968. Vyacheslav Krushkal was supported in part by Simons Foundation fellowship 608604, and NSF grant DMS-2105467.

Acknowledgements

Peter K. Johnson thanks his advisor, Tom Mark, for his continued support and introducing him to lattice cohomology. Vyacheslav Krushkal is grateful to Sergei Gukov for discussions on the GPPV invariant.

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Received: 2021-10-29
Revised: 2022-11-19
Published Online: 2023-02-23
Published in Print: 2023-03-01

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