A non-commutative Reidemeister-Turaev torsion of homology cylinders

We compute the Reidemeister-Turaev torsion of homology cylinders which takes values in the \(K_1\)-group of the \(I\)-adic completion of the group ring \(\mathbb{Q}\pi_1\Sigma_{g,1}\), and prove that its reduction to \(\widehat{\mathbb{Q}\pi_1\Sigma_{g,1}}/\hat{I}^{d+1}\) is a finite-type invariant...

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Bibliographic Details
Published inarXiv.org
Main Authors Nozaki, Yuta, Sato, Masatoshi, Suzuki, Masaaki
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 18.06.2023
Subjects
Cylinders
Homology
Mathematics - Geometric Topology
Mathematics - Quantum Algebra
Rings (mathematics)
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Summary:We compute the Reidemeister-Turaev torsion of homology cylinders which takes values in the \(K_1\)-group of the \(I\)-adic completion of the group ring \(\mathbb{Q}\pi_1\Sigma_{g,1}\), and prove that its reduction to \(\widehat{\mathbb{Q}\pi_1\Sigma_{g,1}}/\hat{I}^{d+1}\) is a finite-type invariant of degree \(d\). We also show that the \(1\)-loop part of the LMO homomorphism and the Enomoto-Satoh trace can be recovered from the leading term of our torsion.
ISSN:2331-8422
DOI:10.48550/arxiv.2206.13019