An introduction to the abelian Reidemeister torsion of three-dimensional manifolds
[Une introduction à la torsion de Reidemeister abélienne des variétés de dimension trois]
Annales Mathématiques Blaise Pascal, Tome 18 (2011) no. 1, pp. 61-140.

Ces notes accompagnent un mini-cours donné à l’école d’automne “Tresses in Pau” en Octobre 2009. Nous introduisons la torsion de Reidemeister des variétés de dimension 3 et ses raffinements par Turaev. Nous survolons quelques applications, et notamment les relations qu’entretient la torsion de Reidemeister avec d’autres invariants classiques.

These notes accompany some lectures given at the autumn school “Tresses in Pau” in October 2009. The abelian Reidemeister torsion for 3-manifolds, and its refinements by Turaev, are introduced. Some applications, including relations between the Reidemeister torsion and other classical invariants, are surveyed.

DOI : https://doi.org/10.5802/ambp.294
Classification : 57M27,  57N10,  57M05,  57M10
Mots clés : variété de dimension 3, torsion de Reidemeister,pôlynome d’Alexander
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     pages = {61--140},
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     doi = {10.5802/ambp.294},
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Gwénaël Massuyeau. An introduction to the abelian Reidemeister torsion of three-dimensional manifolds. Annales Mathématiques Blaise Pascal, Tome 18 (2011) no. 1, pp. 61-140. doi : 10.5802/ambp.294. https://ambp.centre-mersenne.org/articles/10.5802/ambp.294/

[1] James W. Alexander Topological invariants of knots and links, Trans. Amer. Math. Soc., Volume 30 (1928) no. 2, pp. 275-306 | Article | MR 1501429

[2] Reinhold Baer Kurventypen auf Flächen., J. f. M., Volume 156 (1927), pp. 231-246

[3] Reinhold Baer Isotopie von Kurven auf orientierbaren, geschlossenen Flächen und ihr Zusamenhang mit der topologischen Deformation der Flächen., J. f. M., Volume 159 (1928), pp. 101-116

[4] Joan S. Birman Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J., 1974 (Annals of Mathematics Studies, No. 82) | MR 375281 | Zbl 0305.57013

[5] Joan S. Birman; Xiao-Song Lin Knot polynomials and Vassiliev’s invariants, Invent. Math., Volume 111 (1993) no. 2, pp. 225-270 | Article | MR 1198809 | Zbl 0812.57011

[6] Richard C. Blanchfield Applications of free differential calculus to the theory of groups, Senior thesis, Princeton University, 1949

[7] Richard C. Blanchfield Intersection theory of manifolds with operators with applications to knot theory, Ann. of Math. (2), Volume 65 (1957), pp. 340-356 | Article | MR 85512 | Zbl 0080.16601

[8] Francis Bonahon Difféotopies des espaces lenticulaires, Topology, Volume 22 (1983) no. 3, pp. 305-314 | Article | MR 710104 | Zbl 0526.57009

[9] Glen E. Bredon Topology and geometry, Graduate Texts in Mathematics, Volume 139, Springer-Verlag, New York, 1993 | MR 1224675 | Zbl 0934.55001

[10] Thomas A. Chapman Topological invariance of Whitehead torsion, Amer. J. Math., Volume 96 (1974), pp. 488-497 | Article | MR 391109 | Zbl 0358.57004

[11] Tim D. Cochran Noncommutative knot theory, Algebr. Geom. Topol., Volume 4 (2004), pp. 347-398 | Article | MR 2077670 | Zbl 1063.57011

[12] Tim D. Cochran; Amir Gerges; Kent Orr Dehn surgery equivalence relations on 3-manifolds, Math. Proc. Cambridge Philos. Soc., Volume 131 (2001) no. 1, pp. 97-127 | Article | MR 1833077 | Zbl 0984.57010

[13] Marshall M. Cohen A course in simple-homotopy theory, Springer-Verlag, New York, 1973 (Graduate Texts in Mathematics, Vol. 10) | MR 362320 | Zbl 0261.57009

[14] Florian Deloup; Gwénaël Massuyeau Reidemeister-Turaev torsion modulo one of rational homology three-spheres, Geom. Topol., Volume 7 (2003), p. 773-787 (electronic) | Article | MR 2026547 | Zbl 1067.57007

[15] Florian Deloup; Gwénaël Massuyeau Quadratic functions and complex spin structures on three-manifolds, Topology, Volume 44 (2005) no. 3, pp. 509-555 | Article | MR 2122215 | Zbl 1071.57009

[16] Ralph H. Fox Free differential calculus. I. Derivation in the free group ring, Ann. of Math. (2), Volume 57 (1953), pp. 547-560 | Article | MR 53938 | Zbl 0050.25602

[17] Ralph H. Fox Free differential calculus. V. The Alexander matrices re-examined, Ann. of Math. (2), Volume 71 (1960), pp. 408-422 | Article | MR 111781 | Zbl 0142.22305

[18] Wolfgang Franz Über die Torsion einer Überdeckung., J. Reine Angew. Math., Volume 173 (1935), pp. 245-254 | Article | Zbl 0012.12702

[19] Wolfgang Franz Torsionsideale, Torsionsklassen und Torsion., J. Reine Angew. Math., Volume 176 (1936), pp. 113-124 | Zbl 0015.37504

[20] Stefan Friedl Reidemeister torsion, the Thurston norm and Harvey’s invariants, Pacific J. Math., Volume 230 (2007) no. 2, pp. 271-296 | Article | MR 2309160 | Zbl 1163.57008

[21] Stefan Friedl; Taehee Kim The Thurston norm, fibered manifolds and twisted Alexander polynomials, Topology, Volume 45 (2006) no. 6, pp. 929-953 | Article | MR 2263219 | Zbl 1105.57009

[22] Stefan Friedl; Stefano Vidussi A survey of twisted Alexander polynomials, Preprint arXiv:0905.0591, 2009 | MR 2503517

[23] Charles Frohman; Joel Hass Unstable minimal surfaces and Heegaard splittings, Invent. Math., Volume 95 (1989) no. 3, pp. 529-540 | Article | MR 979363 | Zbl 0678.57009

[24] Shelly L. Harvey Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm, Topology, Volume 44 (2005) no. 5, pp. 895-945 | Article | MR 2153977 | Zbl 1080.57019

[25] Jonathan Hillman Algebraic invariants of links, Series on Knots and Everything, Volume 32, World Scientific Publishing Co. Inc., River Edge, NJ, 2002 | MR 1932169 | Zbl 1007.57001

[26] Michael Hutchings; Yi-Jen Lee Circle-valued Morse theory and Reidemeister torsion, Geom. Topol., Volume 3 (1999), p. 369-396 (electronic) | Article | MR 1716272 | Zbl 0929.57019

[27] William Jaco Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, Volume 43, American Mathematical Society, Providence, R.I., 1980 | MR 565450 | Zbl 0433.57001

[28] Akio Kawauchi; Sadayoshi Kojima Algebraic classification of linking pairings on 3-manifolds, Math. Ann., Volume 253 (1980) no. 1, pp. 29-42 | Article | MR 594531 | Zbl 0427.57001

[29] Peter B. Kronheimer; Tomasz S. Mrowka Scalar curvature and the Thurston norm, Math. Res. Lett., Volume 4 (1997) no. 6, pp. 931-937 | MR 1492131 | Zbl 0892.57011

[30] Jean Lannes; François Latour Signature modulo 8 des variétés de dimension 4k dont le bord est stablement parallélisé, C. R. Acad. Sci. Paris Sér. A, Volume 279 (1974), pp. 705-707 | MR 440575 | Zbl 0303.57011

[31] Thang T. Le Finite-type Invariants of 3-manifolds, Encyclopedia of Mathematical Physics (2006), pp. 348-356 | Article

[32] William B. R. Lickorish A representation of orientable combinatorial 3-manifolds, Ann. of Math. (2), Volume 76 (1962), pp. 531-540 | Article | MR 151948 | Zbl 0106.37102

[33] Eduard Looijenga; Jonathan Wahl Quadratic functions and smoothing surface singularities, Topology, Volume 25 (1986) no. 3, pp. 261-291 | Article | MR 842425 | Zbl 0615.32014

[34] Gwénaël Massuyeau Some finiteness properties for the Reidemeister-Turaev torsion of three-manifolds, J. Knot Theory Ramifications, Volume 19 (2010) no. 3, pp. 405-447 | Article | MR 2646639 | Zbl pre05706157

[35] Sergei V. Matveev Generalized surgeries of three-dimensional manifolds and representations of homology spheres, Mat. Zametki, Volume 42 (1987) no. 2, p. 268-278, 345 ((in Russian). English translation: Math. Notes, 42(1-2):651–656, 1987.) | MR 915115 | Zbl 0634.57006

[36] Curtis T. McMullen The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology, Ann. Sci. École Norm. Sup. (4), Volume 35 (2002) no. 2, pp. 153-171 | Numdam | MR 1914929 | Zbl 1009.57021

[37] Guowu Meng; Clifford H. Taubes SW ̲= Milnor torsion, Math. Res. Lett., Volume 3 (1996) no. 5, pp. 661-674 | MR 1418579 | Zbl 0870.57018

[38] John Milnor A duality theorem for Reidemeister torsion, Ann. of Math. (2), Volume 76 (1962), pp. 137-147 | Article | MR 141115 | Zbl 0108.36502

[39] John Milnor Whitehead torsion, Bull. Amer. Math. Soc., Volume 72 (1966), pp. 358-426 | Article | MR 196736 | Zbl 0147.23104

[40] John Milnor Infinite cyclic coverings, Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967), Prindle, Weber & Schmidt, Boston, Mass., 1968, pp. 115-133 | MR 242163 | Zbl 0179.52302

[41] Edwin E. Moise Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung, Ann. of Math. (2), Volume 56 (1952), pp. 96-114 | Article | MR 48805 | Zbl 0048.17102

[42] John W. Morgan; Dennis P. Sullivan The transversality characteristic class and linking cycles in surgery theory, Ann. of Math. (2), Volume 99 (1974), pp. 463-544 | Article | MR 350748 | Zbl 0295.57008

[43] James Munkres Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Ann. of Math. (2), Volume 72 (1960), pp. 521-554 | Article | MR 121804 | Zbl 0108.18101

[44] Hitoshi Murakami A weight system derived from the multivariable Conway potential function, J. London Math. Soc. (2), Volume 59 (1999) no. 2, pp. 698-714 | Article | MR 1709675 | Zbl 0922.57008

[45] Liviu I. Nicolaescu The Reidemeister torsion of 3-manifolds, de Gruyter Studies in Mathematics, Volume 30, Walter de Gruyter & Co., Berlin, 2003 | MR 1968575 | Zbl 1024.57002

[46] Liviu I. Nicolaescu Seiberg-Witten invariants of rational homology 3-spheres, Commun. Contemp. Math., Volume 6 (2004) no. 6, pp. 833-866 | Article | MR 2111431 | Zbl 1082.57010

[47] Tomotada Ohtsuki Finite type invariants of integral homology 3-spheres, J. Knot Theory Ramifications, Volume 5 (1996) no. 1, pp. 101-115 | Article | MR 1373813 | Zbl 0942.57009

[48] Peter Ozsváth; Zoltán Szabó Holomorphic disks and genus bounds, Geom. Topol., Volume 8 (2004), p. 311-334 (electronic) | Article | MR 2023281 | Zbl 1056.57020

[49] Peter Ozsváth; Zoltán Szabó Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. (2), Volume 159 (2004) no. 3, pp. 1159-1245 | Article | MR 2113020 | Zbl 1081.57013

[50] Peter Ozsváth; Zoltán Szabó Link Floer homology and the Thurston norm, J. Amer. Math. Soc., Volume 21 (2008) no. 3, pp. 671-709 | Article | MR 2393424 | Zbl pre05759134

[51] Joan Porti Torsion de Reidemeister pour les variétés hyperboliques, Mem. Amer. Math. Soc., Volume 128 (1997) no. 612, x+139 pages | MR 1396960

[52] Mikhail M. Postnikov The structure of the ring of intersections of three-dimensional manifolds, Doklady Akad. Nauk. SSSR (N.S.), Volume 61 (1948), pp. 795-797 ((in Russian).) | MR 27513 | Zbl 0034.25702

[53] Kurt Reidemeister Homotopieringe und Linsenräume., Abh. Math. Semin. Hamb. Univ., Volume 11 (1935), pp. 102-109 | Article | Zbl 0011.32404

[54] George de Rham; Serge Maumary; Michel A. Kervaire Torsion et type simple d’homotopie, Exposés faits au séminaire de Topologie de l’Université de Lausanne. Lecture Notes in Mathematics, No. 48, Springer-Verlag, Berlin, 1967 | MR 222893 | Zbl 0153.53901

[55] Dale Rolfsen Knots and links, Publish or Perish Inc., Berkeley, Calif., 1976 (Mathematics Lecture Series, No. 7) | MR 515288 | Zbl 0854.57002

[56] Colin P. Rourke; Brian J. Sanderson Introduction to piecewise-linear topology, Springer-Verlag, New York, 1972 (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69) | MR 665919

[57] Hans-Georg Schumann Über Moduln und Gruppenbilder, Math. Ann., Volume 114 (1937) no. 1, pp. 385-413 | Article | MR 1513146 | Zbl 0016.29403

[58] Herbert Seifert; William Threlfall Seifert and Threlfall: a textbook of topology, Pure and Applied Mathematics, Volume 89, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980 | MR 575168

[59] Norman Steenrod The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951 | MR 39258 | Zbl 0054.07103

[60] Dennis Sullivan On the intersection ring of compact three manifolds, Topology, Volume 14 (1975) no. 3, pp. 275-277 | Article | MR 383415 | Zbl 0312.57003

[61] Masaaki Suzuki The Magnus representation of the Torelli group g,1 is not faithful for g2, Proc. Amer. Math. Soc., Volume 130 (2002) no. 3, p. 909-914 (electronic) | Article | MR 1866048 | Zbl 0993.57006

[62] Masaaki Suzuki On the kernel of the Magnus representation of the Torelli group, Proc. Amer. Math. Soc., Volume 133 (2005) no. 6, p. 1865-1872 (electronic) | Article | MR 2120289 | Zbl 1069.57010

[63] Gadde A. Swarup On a theorem of C. B. Thomas, J. London Math. Soc. (2), Volume 8 (1974), pp. 13-21 | Article | MR 341474 | Zbl 0281.57003

[64] Charles B. Thomas The oriented homotopy type of compact 3-manifolds, Proc. London Math. Soc. (3), Volume 19 (1969), pp. 31-44 | Article | MR 248838 | Zbl 0167.21502

[65] William P. Thurston A norm for the homology of 3-manifolds, Mem. Amer. Math. Soc., Volume 59 (1986) no. 339, p. i-vi and 99–130 | MR 823443 | Zbl 0585.57006

[66] William P. Thurston Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, Volume 35, Princeton University Press, Princeton, NJ, 1997 (Edited by Silvio Levy) | MR 1435975 | Zbl 0873.57001

[67] Lorenzo Traldi Milnor’s invariants and the completions of link modules, Trans. Amer. Math. Soc., Volume 284 (1984) no. 1, pp. 401-424 | MR 742432 | Zbl 0525.57004

[68] Lorenzo Traldi Conway’s potential function and its Taylor series, Kobe J. Math., Volume 5 (1988) no. 2, pp. 233-263 | MR 990824 | Zbl 0732.57007

[69] Vladimir Turaev The Alexander polynomial of a three-dimensional manifold, Mat. Sb. (N.S.), Volume 97(139) (1975), p. 341-359, 463 ((in Russian). English translation: Math. USSR Sb., 26:313–329, 1975.) | MR 383425 | Zbl 0315.57001

[70] Vladimir Turaev Reidemeister torsion and the Alexander polynomial, Mat. Sb. (N.S.), Volume 18(66) (1976), pp. 252-270 ((in Russian). English translation: Math. USSR Sb., 30:221–237, 1976.) | MR 433462 | Zbl 0364.57004

[71] Vladimir Turaev Cohomology rings, linking coefficient forms and invariants of spin structures in three-dimensional manifolds, Mat. Sb. (N.S.), Volume 120(162) (1983), p. 68-83, 143 ((in Russian). English translation: Math. USSR Sb., 48:65–79, 1984.) | MR 687337 | Zbl 0519.57009

[72] Vladimir Turaev Reidemeister torsion in knot theory, Uspekhi Mat. Nauk, Volume 41 (1986) no. 1, p. 97-147, 240 ((in Russian). English translation: Russian Math. Surveys, 41:119–182, 1986.) | MR 832411 | Zbl 0602.57005

[73] Vladimir Turaev Euler structures, nonsingular vector fields, and Reidemeister-type torsions, Izv. Akad. Nauk SSSR Ser. Mat., Volume 53 (1989) no. 3, p. 607-643, 672 ((in Russian). English translation: Math. USSR Izvestia, 34:627–662, 1990.) | MR 1013714 | Zbl 0692.57015

[74] Vladimir Turaev Torsion invariants of Spin c -structures on 3-manifolds, Math. Res. Lett., Volume 4 (1997) no. 5, pp. 679-695 | MR 1484699 | Zbl 0891.57019

[75] Vladimir Turaev A combinatorial formulation for the Seiberg-Witten invariants of 3-manifolds, Math. Res. Lett., Volume 5 (1998) no. 5, pp. 583-598 | MR 1666856 | Zbl 1002.57036

[76] Vladimir Turaev Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001 (Notes taken by Felix Schlenk) | MR 1809561 | Zbl 0970.57001

[77] Vladimir Turaev A homological estimate for the Thurston norm, Preprint arXiv:math/0207267, 2002 | MR 1885218

[78] Vladimir Turaev Torsions of 3-dimensional manifolds, Progress in Mathematics, Volume 208, Birkhäuser Verlag, Basel, 2002 | MR 1958479 | Zbl 1012.57002

[79] Charles T. C. Wall Quadratic forms on finite groups, and related topics, Topology, Volume 2 (1963), pp. 281-298 | Article | MR 156890 | Zbl 0215.39903

[80] Andrew H. Wallace Modifications and cobounding manifolds, Canad. J. Math., Volume 12 (1960), pp. 503-528 | Article | MR 125588 | Zbl 0108.36101

[81] J. H. C. Whitehead Manifolds with transverse fields in euclidean space, Ann. of Math. (2), Volume 73 (1961), pp. 154-212 | Article | MR 124917 | Zbl 0096.37802