A non-commutative Reidemeister-Turaev torsion of homology cylinders

Nozaki, Yuta; Sato, Masatoshi; Suzuki, Masaaki

doi:10.1090/tran/8925

A non-commutative Reidemeister-Turaev torsion of homology cylinders
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by Yuta Nozaki, Masatoshi Sato and Masaaki Suzuki;

Trans. Amer. Math. Soc. 376 (2023), 5045-5088

DOI: https://doi.org/10.1090/tran/8925

Published electronically: April 19, 2023

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Abstract:

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.

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