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.

48 pages, 7 figures

References in corpus (5)

• Abelian quotients of the $Y$-filtration on the homology cylinders via the LMO functor
• New series in the Johnson cokernels of the mapping class groups of surfaces
• An introduction to the abelian Reidemeister torsion of three-dimensional manifolds
• On the kernel of the surgery map restricted to the 1-loop part
• Topological formula of the loop expansion of the colored Jones polynomials

Cited by in corpus (1)

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