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 K_1 -group of the I I -adic completion of the group ring Q π 1 Σ g , 1 \mathbb {Q}\pi _1\Sigma _{g,1} , and prove that its reduction to Q π 1 Σ g , 1 ^ / I ^ d + 1 \widehat {\mathbb {Q}\pi _1\Sigma _{g,1}}/\hat {I}^{d+1} is a finite-type invariant of degree d d . We also show that the 1 1 -loop part of the LMO homomorphism and the Enomoto-Satoh trace can be recovered from the leading term of our torsion.

Lagrangian traces for the Johnson filtration of the handlebody group

“…The equality also seems to hold in degree 3, hence the following question: Question 4. 19 In the sequel, a chiral necklace, or chiral tensor, is a cyclic tensor equal to its mirror image (up to cyclic permutation).…”

“…The invariant κ has been recently generalized by Nozaki, Sato, and Suzuki[19], to a more general invariant of homology cylinders vanishing on the mapping class group.It might be interesting to investigate the relation with the Johnson homomorphisms, i.e. to generalize Proposition 6.1, and possibly relate their invariant to Tr.…”