A non-commutative Reidemeister-Turaev torsion of homology cylinders

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.

arXiv comment48 pages, 7 figures

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Cited by in corpus (1)

• Torsion elements in the associated graded of the $Y$-filtration of the monoid of homology cylinders