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A geometric description of the Reidemeister–Turaev torsion for abelian representations on 3-manifolds

Part of: Low-dimensional topology

Published online by Cambridge University Press:  21 May 2025

TATSURO SHIMIZU
TATSURO SHIMIZU*
Affiliation:
Faculty of Policy Management, Keio University, Fujisawa, Kanagawa, Japan e-mail: tshimizu@sfc.keio.ac.jp
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Abstract

Let M be a closed oriented 3-manifold equipped with an Euler structure e and an acyclic representation of its fundamental group. We define a twisted self-linking homology class of the diagonal of the two-point configuration space of M with respect to e. This twisted self-linking homology class appears as an obstruction in the Chern–Simons perturbation theory. When the representation is the maximal free abelian representation $\rho_0$, we prove that our self-linking class is a properly defined “logarithmic derivative” of the Reidemeister–Turaev torsion of $(M,\rho_0,e)$ equipped with the given Euler structure.

MSC classification

Secondary: 57M05: Fundamental group, presentations, free differential calculus
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 26
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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