A Legendrian Turaev torsion via generating families
Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 57-119.

We introduce a Legendrian invariant built out of the Turaev torsion of generating families. This invariant is defined for a certain class of Legendrian submanifolds of 1-jet spaces, which we call of Euler type. We use our invariant to study mesh Legendrians: a family of 2-dimensional Euler type Legendrian links whose linking pattern is determined by a bicolored trivalent ribbon graph. The Turaev torsion of mesh Legendrians is related to a certain monodromy of handle slides, which we compute in terms of the combinatorics of the graph. As an application, we exhibit pairs of Legendrian links in the 1-jet space of any orientable closed surface which are formally equivalent, cannot be distinguished by any natural Legendrian invariant, yet are not Legendrian isotopic. These examples appeared in a different guise in the work of the second author with J. Klein on pictures for K 3 and the higher Reidemeister torsion of circle bundles.

Nous introduisons un invariant des sous-variétés legendriennes construit à l’aide de fonctions génératrices. Cet invariant est défini pour une certaine classe de sous-variétés legendriennes, que nous appelons de type d’Euler, dans un espace de 1-jets. Nous utilisons cet invariant pour étudier les mailles legendriennes : une famille de sous-variétés legendriennes de type d’Euler dont le motif d’entrelac est déterminé par un graphe bicolore et trivalent qui est muni d’un ordre cyclique des arêtes concourantes en un même sommet. La torsion de Turaev d’une maille legendrienne est reliée à une certaine monodromie de glissement d’anses, que nous calculons en terme de la combinatoire du graphe. Comme application, nous exhibons, dans l’espace des 1-jets de toute surface fermée orientable, des paires d’entrelacs legendriens qui sont formellement équivalents, ne peuvent être distingués par aucun invariant legendrien naturel, et pourtant ne sont pas isotopes parmi les variétés legendriennes. Ces exemples sont apparus sous une forme différente dans les travaux du second auteur avec J. Klein sur des dessins pour K 3 et sur la torsion de Reidemeister supérieure de fibrés en cercles.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.141
Classification: 57R17,  19J10
Keywords: Legendrians, Turaev torsion, K-theory
Daniel Álvarez-Gavela 1; Kiyoshi Igusa 2

1 Department of Mathematics, Princeton University Princeton, NJ 086540, USA
2 Department of Mathematics, Brandeis University PO Box 9110, Waltham, MA 02454-9110, USA
@article{JEP_2021__8__57_0,
     author = {Daniel \'Alvarez-Gavela and Kiyoshi Igusa},
     title = {A {Legendrian} {Turaev} torsion via generating~families},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {57--119},
     publisher = {\'Ecole polytechnique},
     volume = {8},
     year = {2021},
     doi = {10.5802/jep.141},
     mrnumber = {4180260},
     zbl = {07282222},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.141/}
}
TY  - JOUR
TI  - A Legendrian Turaev torsion via generating families
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2021
DA  - 2021///
SP  - 57
EP  - 119
VL  - 8
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.141/
UR  - https://www.ams.org/mathscinet-getitem?mr=4180260
UR  - https://zbmath.org/?q=an%3A07282222
UR  - https://doi.org/10.5802/jep.141
DO  - 10.5802/jep.141
LA  - en
ID  - JEP_2021__8__57_0
ER  - 
%0 Journal Article
%T A Legendrian Turaev torsion via generating families
%J Journal de l’École polytechnique — Mathématiques
%D 2021
%P 57-119
%V 8
%I École polytechnique
%U https://doi.org/10.5802/jep.141
%R 10.5802/jep.141
%G en
%F JEP_2021__8__57_0
Daniel Álvarez-Gavela; Kiyoshi Igusa. A Legendrian Turaev torsion via generating families. Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 57-119. doi : 10.5802/jep.141. https://jep.centre-mersenne.org/articles/10.5802/jep.141/

[AGZV12] V. I. Arnold, S. M. Gusein-Zade & A. N. Varchenko - Singularities of differentiable maps. Volume 1, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2012 | Zbl: 1290.58001

[AK18] M. Abouzaid & T. Kragh - “Simple homotopy equivalence of nearby Lagrangians”, Acta Math. 220 (2018) no. 2, p. 207-237 | Article | MR: 3849284 | Zbl: 1396.53104

[Bar64] D. Barden - On the structure and classification of differential manifolds, Ph. D. Thesis, Cambridge University, 1964

[BFG + 18] K. Baur, E. Faber, S. Gratz, K. Serhiyenko & G. Todorov - “Friezes satisfying higher SL k -determinants”, 2018, to appear in Algebra & Number Theory | 1810.10562

[BL95] J.-M. Bismut & J. Lott - “Flat vector bundles, direct images and higher real analytic torsion”, J. Amer. Math. Soc. 8 (1995) no. 2, p. 291-363 | Article | MR: 1303026 | Zbl: 0837.58028

[Cer70] J. Cerf - “La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie”, Publ. Math. Inst. Hautes Études Sci. 39 (1970), p. 5-173 | Numdam | MR: 292089 | Zbl: 0213.25202

[Cha84] M. Chaperon - “Une idée du type ‘géodésiques brisées’ pour les systèmes hamiltoniens”, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984) no. 13, p. 293-296 | Zbl: 0576.58010

[Cha19] F. Charette - “Quantum Reidemeister torsion, open Gromov-Witten invariants and a spectral sequence of Oh”, Internat. Math. Res. Notices (2019) no. 8, p. 2483-2518 | Article | MR: 3942168 | Zbl: 1431.83064

[Che96] Y. V. Chekanov - “Critical points of quasi-functions and generating families of Legendrian manifolds”, Funct. Anal. Appl. 30 (1996) no. 2, p. 118-128 | Article | Zbl: 0873.58017

[Che02] Y. V. Chekanov - “Differential algebra of Legendrian links”, Invent. Math. 150 (2002) no. 3, p. 441-483 | Article | MR: 1946550 | Zbl: 1029.57011

[CM18] R. Casals & E. Murphy - “Differential algebra of cubic planar graphs”, Adv. Math. 338 (2018), p. 401-446 | Article | MR: 3861709 | Zbl: 1397.05042

[dR40] G. de Rham - “Sur les complexes avec automorphismes”, Comment. Math. Helv. 12 (1940), p. 191-211 | Article | MR: 2551 | Zbl: 0022.40802

[DR11] G. Dimitroglou Rizell - “Knotted Legendrian surfaces with few Reeb chords”, Algebraic Geom. Topol. 11 (2011) no. 5, p. 2903-2936 | Article | MR: 2846915 | Zbl: 1248.53073

[DWW03] W. Dwyer, M. Weiss & B. Williams - “A parametrized index theorem for the algebraic K-theory Euler class”, Acta Math. 190 (2003) no. 1, p. 1-104 | Article | MR: 1982793 | Zbl: 1077.19002

[EES07] T. Ekholm, J. Etnyre & M. G. Sullivan - “Legendrian contact homology in P×, Trans. Amer. Math. Soc. 359 (2007) no. 7, p. 3301-3335 | Article | MR: 2299457 | Zbl: 1119.53051

[EG98] Y. M. Eliashberg & M. Gromov - “Lagrangian intersection theory: finite-dimensional approach”, in Geometry of differential equations, Amer. Math. Soc. Transl. Ser. 2, vol. 186, American Mathematical Society, Providence, RI, 1998, p. 27-118 | Article | MR: 1732407 | Zbl: 0919.58015

[Eli98] Y. M. Eliashberg - “Invariants in contact topology”, in Proceedings of the ICM, Vol. II (Berlin, 1998), Doc. Math., Deutsche Mathematiker-Vereinigung, Berlin, 1998, p. 327-338, Extra Vol. II | Zbl: 0913.53010

[EM12] Y. M. Eliashberg & N. M. Mishachev - “The space of framed functions is contractible”, in Essays in mathematics and its applications, Springer, Heidelberg, 2012, p. 81-109 | Article | Zbl: 1334.57030

[FI04] D. Fuchs & T. Ishkhanov - “Invariants of Legendrian knots and decompositions of front diagrams”, Moscow Math. J. 4 (2004) no. 3, p. 707-717, 783 | Article | MR: 2119145 | Zbl: 1073.53106

[FR11] D. Fuchs & D. Rutherford - “Generating families and Legendrian contact homology in the standard contact space”, J. Topology 4 (2011) no. 1, p. 190-226 | Article | MR: 2783382 | Zbl: 1237.57026

[Fra35] W. Franz - “Über die Torsion einer Überdeckung”, J. reine angew. Math. 173 (1935), p. 245-254 | Article | Zbl: 0012.12702

[Fuk97] K. Fukaya - “The symplectic s-cobordism conjecture: a summary”, in Geometry and physics (Aarhus, 1995), Lecture Notes in Pure and Appl. Math., vol. 184, Dekker, New York, 1997, p. 209-219 | MR: 1423167 | Zbl: 0871.57032

[GKS12] S. Guillermou, M. Kashiwara & P. Schapira - “Sheaf quantization of Hamiltonian isotopies and applications to nondisplaceability problems”, Duke Math. J. 161 (2012) no. 2, p. 201-245 | Article | MR: 2876930 | Zbl: 1242.53108

[Hen11] M. B. Henry - “Connections between Floer-type invariants and Morse-type invariants of Legendrian knots”, Pacific J. Math. 249 (2011) no. 1, p. 77-133 | Article | MR: 2764942 | Zbl: 1233.57006

[HI19] E. J. Hanson & K. Igusa - “A counterexample to the ϕ-dimension conjecture”, 2019 | 1911.00614

[HL99] M. Hutchings & Y.-J. Lee - “Circle-valued Morse theory and Reidemeister torsion”, Geom. Topol. 3 (1999), p. 369-396 | Article | MR: 1716272 | Zbl: 0929.57019

[HR15] M. B. Henry & D. Rutherford - “Equivalence classes of augmentations and Morse complex sequences of Legendrian knots”, Algebraic Geom. Topol. 15 (2015) no. 6, p. 3323-3353 | Article | MR: 3450763 | Zbl: 1334.57025

[HW73] A. Hatcher & J. Wagoner - Pseudo-isotopies of compact manifolds, Astérisque, vol. 6, Société Mathématique de France, Paris, 1973 | Numdam | MR: 353337 | Zbl: 1384.57019

[Igu] K. Igusa - “The generalized Grassmann invariant”, preprint

[Igu79] K. Igusa - The Wh 3 (π) obstruction for pseudoisotopy, Ph. D. Thesis, Princeton University, 1979 | MR: 2628331

[Igu84] K. Igusa - “What happens to Hatcher and Wagoner’s formulas for π 0 C(M) when the first Postnikov invariant of M is nontrivial?”, in Algebraic K-theory, number theory, geometry and analysis (Bielefeld, 1982), Lect. Notes in Math., vol. 1046, Springer, Berlin, 1984, p. 104-172 | Article | MR: 750679 | Zbl: 0546.57015

[Igu87] K. Igusa - “The space of framed functions”, Trans. Amer. Math. Soc. 301 (1987) no. 2, p. 431-477 | Article | MR: 882699 | Zbl: 0624.57026

[Igu88] K. Igusa - “The stability theorem for smooth pseudoisotopies”, K-Theory 2 (1988) no. 1-2, p. 1-355 | Article | MR: 972368 | Zbl: 0691.57011

[Igu93] K. Igusa - “The Borel regulator map on pictures. I. A dilogarithm formula”, K-Theory 7 (1993) no. 3, p. 201-224 | Article | MR: 1244001 | Zbl: 0793.19001

[Igu02] K. Igusa - Higher Franz-Reidemeister torsion, AMS/IP Studies in Advanced Math., vol. 31, American Mathematical Society, Providence, RI, 2002 | Article | MR: 1945530 | Zbl: 1016.19001

[Igu04] K. Igusa - “Combinatorial Miller-Morita-Mumford classes and Witten cycles”, Algebraic Geom. Topol. 4 (2004), p. 473-520 | Article | MR: 2077674 | Zbl: 1072.57013

[Igu05] K. Igusa - Higher complex torsion and the framing principle, Mem. Amer. Math. Soc., vol. 177, no. 835, American Mathematical Society, Providence, RI, 2005 | Article | Zbl: 1083.57030

[IK93] K. Igusa & J. Klein - “The Borel regulator map on pictures. II. An example from Morse theory”, K-Theory 7 (1993) no. 3, p. 225-267 | Article | MR: 1244002 | Zbl: 0793.19002

[Jek89] S. M. Jekel - “A simplicial formula and bound for the Euler class”, Israel J. Math. 66 (1989) no. 1-3, p. 247-259 | Article | MR: 1017165 | Zbl: 0686.57012

[JKS16] B. T. Jensen, A. D. King & X. Su - “A categorification of Grassmannian cluster algebras”, Proc. London Math. Soc. (3) 113 (2016) no. 2, p. 185-212 | Article | MR: 3534971 | Zbl: 1375.13033

[JT06] J. Jordan & L. Traynor - “Generating family invariants for Legendrian links of unknots”, Algebraic Geom. Topol. 6 (2006), p. 895-933 | Article | MR: 2240920 | Zbl: 1130.57018

[Kle89] J. R. Klein - The cell complex construction and higher R-torsion for bundles with framed Morse functions, Ph. D. Thesis, Brandeis University, 1989 | MR: 2637458

[Kon92] M. Kontsevich - “Intersection theory on the moduli space of curves and the matrix Airy function”, Comm. Math. Phys. 147 (1992) no. 1, p. 1-23 | Article | MR: 1171758 | Zbl: 0756.35081

[Kra18] T. Kragh - “Generating families for Lagrangians in 2n and the Hatcher-Waldhausen map”, 2018 | 1804.02557

[Lau12] F. Laudenbach - Transversalité, courants et théorie de Morse, Éditions de l’École polytechnique, Palaiseau, 2012 | Zbl: 1280.57001

[Lee05a] Y.-J. Lee - “Reidemeister torsion in Floer-Novikov theory and counting pseudo-holomorphic tori. I”, J. Symplectic Geom. 3 (2005) no. 2, p. 221-311 | Article | MR: 2199540 | Zbl: 1093.53091

[Lee05b] Y.-J. Lee - “Reidemeister torsion in Floer-Novikov theory and counting pseudo-holomorphic tori. II”, J. Symplectic Geom. 3 (2005) no. 3, p. 385-480 | Article | MR: 2198782 | Zbl: 1093.53092

[LS85] F. Laudenbach & J.-C. Sikorav - “Persistance d’intersection avec la section nulle au cours d’une isotopie hamiltonienne dans un fibré cotangent”, Invent. Math. 82 (1985) no. 2, p. 349-357 | Article | Zbl: 0592.58023

[Maz63] B. Mazur - “Relative neighborhoods and the theorems of Smale”, Ann. of Math. (2) 77 (1963), p. 232-249 | Article | MR: 150786 | Zbl: 0112.38301

[Mil61] J. Milnor - “Two complexes which are homeomorphic but combinatorially distinct”, Ann. of Math. (2) 74 (1961), p. 575-590 | Article | MR: 133127 | Zbl: 0102.38103

[Mil66] J. Milnor - “Whitehead torsion”, Bull. Amer. Math. Soc. 72 (1966), p. 358-426 | Article | MR: 196736 | Zbl: 0147.23104

[MT96] G. Meng & C. H. Taubes - “SW ̲= Milnor torsion”, Math. Res. Lett. 3 (1996) no. 5, p. 661-674 | Article | MR: 1418579 | Zbl: 0870.57018

[Mur19] E. Murphy - “Loose Legendrian embeddings in high dimensional contact manifolds”, 2019 | 1201.2245v5

[Rei35] K. Reidemeister - “Homotopieringe und Linsenräume”, Abh. Math. Sem. Univ. Hamburg 11 (1935) no. 1, p. 102-109 | Article | MR: 3069647 | Zbl: 0011.32404

[RS71] D. B. Ray & I. M. Singer - “R-torsion and the Laplacian on Riemannian manifolds”, Adv. Math. 7 (1971), p. 145-210 | Article | MR: 295381 | Zbl: 0239.58014

[RS18] D. Rutherford & M. G. Sullivan - “Generating families and augmentations for Legendrian surfaces”, Algebraic Geom. Topol. 18 (2018) no. 3, p. 1675-1731 | Article | MR: 3784016 | Zbl: 1388.53096

[Sab05] J. M. Sabloff - “Augmentations and rulings of Legendrian knots”, Internat. Math. Res. Notices (2005) no. 19, p. 1157-1180 | Article | MR: 2147057 | Zbl: 1082.57020

[Sab06] J. M. Sabloff - “Duality for Legendrian contact homology”, Geom. Topol. 10 (2006), p. 2351-2381 | Article | MR: 2284060 | Zbl: 1128.57026

[Sik86] J.-C. Sikorav - “Sur les immersions lagrangiennes dans un fibré cotangent admettant une phase génératrice globale”, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986) no. 3, p. 119-122 | MR: 830282 | Zbl: 0602.58019

[Sma61] S. Smale - “Generalized Poincaré’s conjecture in dimensions greater than four”, Ann. of Math. (2) 74 (1961), p. 391-406 | Article | MR: 137124 | Zbl: 0099.39202

[SS16] J. M. Sabloff & M. G. Sullivan - “Families of Legendrian submanifolds via generating families”, Quantum Topol. 7 (2016) no. 4, p. 639-668 | Article | MR: 3593565 | Zbl: 1358.57029

[STWZ19] V. Shende, D. Treumann, H. Williams & E. Zaslow - “Cluster varieties from Legendrian knots”, Duke Math. J. 168 (2019) no. 15, p. 2801-2871 | Article | MR: 4017516 | Zbl: 07145322

[STZ17] V. Shende, D. Treumann & E. Zaslow - “Legendrian knots and constructible sheaves”, Invent. Math. 207 (2017) no. 3, p. 1031-1133 | Article | MR: 3608288 | Zbl: 1369.57016

[Sul02] M. G. Sullivan - “K-theoretic invariants for Floer homology”, Geom. Funct. Anal. 12 (2002) no. 4, p. 810-872 | Article | MR: 1935550 | Zbl: 1081.53076

[Suá17] L. S. Suárez - “Exact Lagrangian cobordism and pseudo-isotopy”, Internat. J. Math. 28 (2017) no. 8, p. 1750059, 35 | Article | MR: 3681121 | Zbl: 1379.53095

[Tra01] L. Traynor - “Generating function polynomials for Legendrian links”, Geom. Topol. 5 (2001), p. 719-760 | Article | MR: 1871403 | Zbl: 1030.53086

[Tur86] V. G. Turaev - “Reidemeister torsion in knot theory”, Uspehi Mat. Nauk 41 (1986) no. 1(247), p. 97-147, 240 | MR: 832411 | Zbl: 0602.57005

[Tur98] V. G. Turaev - “A combinatorial formulation for the Seiberg-Witten invariants of 3-manifolds”, Math. Res. Lett. 5 (1998) no. 5, p. 583-598 | Article | MR: 1666856 | Zbl: 1002.57036

[Vit92] C. Viterbo - “Symplectic topology as the geometry of generating functions”, Math. Ann. 292 (1992) no. 4, p. 685-710 | Article | MR: 1157321 | Zbl: 0735.58019

[Wag78] J. B. Wagoner - “Diffeomorphisms, K 2 , and analytic torsion”, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1 (Proc. Sympos. Pure Math.) XXXII (1978), p. 23-33 | Zbl: 0408.57015

[Wal82] F. Waldhausen - “Algebraic K-theory of spaces, a manifold approach”, in Current trends in algebraic topology, Part 1 (London, Ont., 1981), CMS Conf. Proc., vol. 2, American Mathematical Society, Providence, RI, 1982, p. 141-184 | MR: 686115 | Zbl: 0595.57026

[Whi50] J. H. C. Whitehead - “Simple homotopy types”, Amer. J. Math. 72 (1950), p. 1-57 | Article | MR: 35437 | Zbl: 0040.38901

Cited by Sources: