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Reidemeister-Turaev torsion modulo one of rational homology threespheres. (English) Zbl 1067.57007

Given a rational homology three-sphere \(M\) (that is, the homology groups of \(M\) are the same as those of the three-sphere), and a \(\text{Spin}^{c}\)-structure \(\sigma\) on it, one can define two quadratic functions over the linking pairing of \(M\). One quadratic function can be defined by using the Reidemeister–Turaev torsion of the pair \((M,\sigma)\), and the other by using the intersection pairing of a simply-connected \(\text{Spin}^{c}\) four-manifold bounding \((M,\sigma)\).
The main result of the paper is to show that these two are equal, which has already been proved in [Liviu I. Nicolaescu, The Reidemeister torsion of 3-manifolds (de Gruyter Studies in Mathematics 30, Walter de Gruyter & Co., Berlin) (2003; Zbl 1024.57002)] in a different way. The proof here is as follows. First write down the formulas of the two quadratic functions for a rational homology three-sphere obtained from the three-sphere by integral Dehn surgery along an algebraically split link (a link whose linking numbers vanish). Then the authors use Ohtsuki’s diagonalizing lemma [T. Ohtsuki, Invent. Math. 123, No. 2, 241–257 (1996; Zbl 0855.57016)].

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
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References:

[1] M F Atiyah, Riemann surfaces and spin structures, Ann. Sci. École Norm. Sup. \((4)\) 4 (1971) 47 · Zbl 0212.56402
[2] F Deloup, On abelian quantum invariants of links in 3-manifolds, Math. Ann. 319 (2001) 759 · Zbl 0984.57003 · doi:10.1007/PL00004458
[3] F Deloup, G Massuyeau, Quadratic functions and complex spin structures on three-manifolds, Topology 44 (2005) 509 · Zbl 1071.57009 · doi:10.1016/j.top.2004.11.003
[4] C Gille, Sur certains invariants récents en topologie de dimension 3, Thèse de Doctorat, Université de Nantes (1998)
[5] F Hirzebruch, A Riemann-Roch theorem for differentiable manifolds, Séminaire Bourbaki 177, Soc. Math. France (1995) 129
[6] D Johnson, Spin structures and quadratic forms on surfaces, J. London Math. Soc. \((2)\) 22 (1980) 365 · Zbl 0454.57011 · doi:10.1112/jlms/s2-22.2.365
[7] R C Kirby, The topology of 4-manifolds, Lecture Notes in Mathematics 1374, Springer (1989) · Zbl 0668.57001 · doi:10.1007/BFb0089031
[8] H B Lawson Jr., M L Michelsohn, Spin geometry, Princeton Mathematical Series 38, Princeton University Press (1989) · Zbl 0688.57001
[9] C Lescop, Global surgery formula for the Casson-Walker invariant, Annals of Mathematics Studies 140, Princeton University Press (1996) · Zbl 0949.57008
[10] E Looijenga, J Wahl, Quadratic functions and smoothing surface singularities, Topology 25 (1986) 261 · Zbl 0615.32014 · doi:10.1016/0040-9383(86)90044-3
[11] L I Nicolaescu, The Reidemeister torsion of 3-manifolds, de Gruyter Studies in Mathematics 30, Walter de Gruyter & Co. (2003) · Zbl 1024.57002
[12] H Seifert, W Threlfall, A Textbook of Topology, Pure and Applied Mathematics 89, Academic Press (1980) · Zbl 0469.55001
[13] V G Turaev, Reidemeister torsion and the Alexander polynomial, Mat. Sb. \((\)N.S.\()\) 18(66) (1976) 252 · Zbl 0381.57003
[14] V G Turaev, Torsion invariants of 3-manifolds (editors J Bryden, F Deloup, P Zvengrowski), PIMS Distinguished Chair Lectures, University of Calgary (2001)
[15] V G Turaev, Euler structures, nonsingular vector fields, and Reidemeister-type torsions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989) 607, 672 · Zbl 0692.57015
[16] V Turaev, Torsion invariants of \(\mathrm{Spin}^c\)-structures on 3-manifolds, Math. Res. Lett. 4 (1997) 679 · Zbl 0891.57019 · doi:10.4310/MRL.1997.v4.n5.a6
[17] V Turaev, Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag (2001) · Zbl 0970.57001
[18] V G Turaev, Surgery formula for torsions and Seiberg-Witten invariants of 3-manifolds,
[19] V Turaev, Torsions of 3-dimensional manifolds, Progress in Mathematics 208, Birkhäuser Verlag (2002) · Zbl 1012.57002
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