Wakijo, Naoko Twisted Reidemeister torsions via Heegaard splittings. (English) Zbl 1473.57054 Topology Appl. 299, Article ID 107731, 22 p. (2021). The Reidemeister torsion is a combinatorial invariant of homological complexes, which is defined as an alternating product of determinants. It follows from a deep theorem of T. A. Chapman [Am. J. Math. 96, 488–497 (1974; Zbl 0358.57004)] that it is a topological invariant of manifolds: it does not depend on the choice of the simplicial or cellular decomposition. However, it is not a homotopy invariant, and it has been celebrated for distinguishing between some homotopic but non-homeomorphic 3-manifolds, called lens spaces (see [K. Reidemeister, Abh. Math. Semin. Univ. Hamb. 11, 102–109 (1935; Zbl 0011.32404; JFM 61.1352.01)]).It is in general a difficult task to compute this invariant. In the case of a knot complement, there is essentially one determinant to compute. The relevant matrix can be obtained by Fox calculus, once a special presentation of the fundamental group of the knot is given.The present paper proposes a combinatorial framework to compute the Reidemeister torsion of a closed 3-manifold from a Heegard splitting. The author uses results of A. J. Sieradski [Invent. Math. 84, 121–139 (1986; Zbl 0604.57001)] to describe one of the determinants involved in the computation of the torsion for the homological complex given by the Heegard splitting. The other determinants can be computed as for knots.As it is mentioned in the paper, for a given manifold it is not easy to get an explicit expression for this determinant. Nevertheless computations are performed for some Seifert fibered 3-manifolds, where the author recovers the results obtained by T. Kitano in [Tokyo J. Math. 17, No. 1, 59–75 (1994; Zbl 0846.55010)]. Reviewer: Léo Bénard (Göttingen) Cited in 1 Document MSC: 57K30 General topology of 3-manifolds 57K31 Invariants of 3-manifolds (including skein modules, character varieties) 57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. 57M05 Fundamental group, presentations, free differential calculus Keywords:Reidemeister torsion; 3-manifolds; homology with local coefficients Citations:Zbl 0358.57004; Zbl 0011.32404; Zbl 0604.57001; Zbl 0846.55010; JFM 61.1352.01 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Fox, R. H., Free differential calculus. I: Derivation in the free group ring, Ann. Math. (2), 57, 547-560 (1953) · Zbl 0050.25602 [2] Hog-Angeloni, C.; Metzler, W.; Sieradski, A. J., Two-dimensional Homotopy and Combinatorial Group Theory, London Math. Soc. Lecture Note Series, vol. 197 (1993), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0788.00031 [3] D. 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