An introduction to the abelian Reidemeister torsion of three-dimensional manifolds

Massuyeau, Gwénaël

doi:10.5802/ambp.294

An introduction to the abelian Reidemeister torsion of three-dimensional manifolds [ Une introduction à la torsion de Reidemeister abélienne des variétés de dimension trois ]

Massuyeau, Gwénaël

Annales Mathématiques Blaise Pascal, Tome 18 (2011) no. 1, pp. 61-140.

Résumé
Abstract

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.

MR 2830089 | Zbl pre05903954

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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     title = {An introduction to the abelian Reidemeister torsion of three-dimensional manifolds},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {61--140},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {18},
     number = {1},
     year = {2011},
     doi = {10.5802/ambp.294},
     mrnumber = {2830089},
     zbl = {pre05903954},
     language = {en},
     url = {www.numdam.org/item/AMBP_2011__18_1_61_0/}
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Massuyeau, Gwénaël. 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. http://www.numdam.org/item/AMBP_2011__18_1_61_0/

Bibliographie

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

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

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

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

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

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

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

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

[12] Cochran, Tim D.; Gerges, Amir; Orr, Kent 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] Cohen, Marshall M. A course in simple-homotopy theory, Springer-Verlag, New York, 1973 (Graduate Texts in Mathematics, Vol. 10) | MR 362320 | Zbl 0261.57009

[14] Deloup, Florian; Massuyeau, Gwénaël 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] Deloup, Florian; Massuyeau, Gwénaël 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] Fox, Ralph H. 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] Fox, Ralph H. 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] Franz, Wolfgang Über die Torsion einer Überdeckung., J. Reine Angew. Math., Volume 173 (1935), pp. 245-254 | Article | Zbl 0012.12702

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

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

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

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

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

[30] Lannes, Jean; Latour, François Signature modulo $8$ des variétés de dimension $4 k$ 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] Le, Thang T. Finite-type Invariants of $3$ -manifolds, Encyclopedia of Mathematical Physics (2006), pp. 348-356 | Article

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

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

[34] Massuyeau, Gwénaël 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] Matveev, Sergei V. 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] McMullen, Curtis T. 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] Meng, Guowu; Taubes, Clifford H. $\underset{̲}{SW} =$ Milnor torsion, Math. Res. Lett., Volume 3 (1996) no. 5, pp. 661-674 | MR 1418579 | Zbl 0870.57018

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

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

[40] Milnor, John 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] Moise, Edwin E. 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] Morgan, John W.; Sullivan, Dennis P. 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] Munkres, James 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] Murakami, Hitoshi 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] Nicolaescu, Liviu I. 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] Nicolaescu, Liviu I. 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] Ohtsuki, Tomotada 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] Ozsváth, Peter; Szabó, Zoltán Holomorphic disks and genus bounds, Geom. Topol., Volume 8 (2004), p. 311-334 (electronic) | Article | MR 2023281 | Zbl 1056.57020

[49] Ozsváth, Peter; Szabó, Zoltán 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] Ozsváth, Peter; Szabó, Zoltán 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] Porti, Joan Torsion de Reidemeister pour les variétés hyperboliques, Mem. Amer. Math. Soc., Volume 128 (1997) no. 612, pp. x+139 | MR 1396960

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

[54] de Rham, George; Maumary, Serge; Kervaire, Michel A. 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] Rolfsen, Dale Knots and links, Publish or Perish Inc., Berkeley, Calif., 1976 (Mathematics Lecture Series, No. 7) | MR 515288 | Zbl 0854.57002

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

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

[58] Seifert, Herbert; Threlfall, William 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] Steenrod, Norman The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951 | MR 39258 | Zbl 0054.07103

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

[61] Suzuki, Masaaki The Magnus representation of the Torelli group $ℐ_{g, 1}$ is not faithful for $g \geq 2$ , Proc. Amer. Math. Soc., Volume 130 (2002) no. 3, p. 909-914 (electronic) | Article | MR 1866048 | Zbl 0993.57006

[62] Suzuki, Masaaki 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] Swarup, Gadde A. 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] Thomas, Charles B. 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] Thurston, William P. 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] Thurston, William P. 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] Traldi, Lorenzo 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] Traldi, Lorenzo 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] Turaev, Vladimir 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] Turaev, Vladimir 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] Turaev, Vladimir 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] Turaev, Vladimir 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] Turaev, Vladimir 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] Turaev, Vladimir 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] Turaev, Vladimir 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] Turaev, Vladimir Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001 (Notes taken by Felix Schlenk) | MR 1809561 | Zbl 0970.57001

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

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

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

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

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