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 ¹

¹ IRMA Université de Strasbourg & CNRS 7 rue René Descartes 67084 Strasbourg France

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.

DOI : 10.5802/ambp.294

Classification : 57M27, 57N10, 57M05, 57M10
Keywords: $3$-manifold, Reidemeister torsion, Alexander polynomial
Mot clés : variété de dimension $3$, torsion de Reidemeister, pôlynome d’Alexander

Affiliations des auteurs :

Massuyeau, Gwénaël ¹

¹ IRMA Université de Strasbourg & CNRS 7 rue René Descartes 67084 Strasbourg France

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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/articles/10.5802/ambp.294/

Bibliographie
Cité par

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

[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 | Zbl

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

[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 | DOI | MR | Zbl

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

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

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

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

[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 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

[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 | DOI | MR | Zbl

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

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

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

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

[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 | Zbl

[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 | Zbl

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

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

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

[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 | DOI | MR

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

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

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

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

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

[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 | DOI | MR | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

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

[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

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

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

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

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

[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) | DOI | MR | Zbl

[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) | DOI | MR | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

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

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

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

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

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