Toward a topological description of Legendrian contact homology of unit conormal bundles

Okamoto, Yukihiro¹

¹Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan, Department of Mathematical Sciences, Tokyo Metropolitan University, Tokyo, Japan

Algebraic and Geometric Topology, Tome 25 (2025) no. 2, pp. 951-1027

Résumé

For a smooth compact submanifold K of a Riemannian manifold Q, its unit conormal bundle ΛK is a Legendrian submanifold of the unit cotangent bundle of Q with a canonical contact structure. Using pseudoholomorphic curve techniques, the Legendrian contact homology of ΛK is defined when, for instance, Q = ℝn. Aiming at giving another description of this homology, we define a graded ℝ-algebra for any pair (Q,K) with orientations from a perspective of string topology and prove its invariance under smooth isotopies of K. We conjecture that it is isomorphic to the Legendrian contact homology of ΛK with coefficients in ℝ in all degrees. This is a reformulation of a homology group, called string homology, introduced by Cieliebak, Ekholm, Latschev and Ng when the codimension of K is 2, though the coefficient is reduced from the original ℤ[π1(ΛK)] to ℝ. We compute our invariant (i) in all degrees for specific examples, and (ii) in the 0 th degree when the normal bundle of K is a trivial 2-plane bundle.

Détail

DOI : 10.2140/agt.2025.25.951

Keywords: string topology, embedded submanifolds, Legendrian contact homology

Bibliographie

[1] A Abbondandolo, A Portaluri, M Schwarz, The homology of path spaces and Floer homology with conormal boundary conditions, J. Fixed Point Theory Appl. 4 (2008) 263 | DOI

[2] A Abbondandolo, M Schwarz, On the Floer homology of cotangent bundles, Comm. Pure Appl. Math. 59 (2006) 254 | DOI

[3] M Abouzaid, Symplectic cohomology and Viterbo’s theorem, from: "Free loop spaces in geometry and topology", IRMA Lect. Math. Theor. Phys. 24, Eur. Math. Soc. (2015) 271

[4] Y Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002) 441 | DOI

[5] K T Chen, On differentiable spaces, from: "Categories in continuum physics", Lecture Notes in Math. 1174, Springer (1986) 38 | DOI

[6] K Cieliebak, T Ekholm, J Latschev, L Ng, Knot contact homology, string topology, and the cord algebra, J. Éc. Polytech. Math. 4 (2017) 661 | DOI

[7] G Dimitroglou Rizell, Lifting pseudo-holomorphic polygons to the symplectisation of P × R and applications, Quantum Topol. 7 (2016) 29 | DOI

[8] T Ekholm, J B Etnyre, L Ng, M G Sullivan, Knot contact homology, Geom. Topol. 17 (2013) 975 | DOI

[9] T Ekholm, J Etnyre, M Sullivan, The contact homology of Legendrian submanifolds in R2n+1, J. Differential Geom. 71 (2005) 177

[10] T Ekholm, J Etnyre, M Sullivan, Orientations in Legendrian contact homology and exact Lagrangian immersions, Int. J. Math. 16 (2005) 453 | DOI

[11] T Ekholm, J Etnyre, M Sullivan, Legendrian contact homology in P × R, Trans. Amer. Math. Soc. 359 (2007) 3301 | DOI

[12] Y Eliashberg, Invariants in contact topology, from: "Proceedings of the International Congress of Mathematicians, II" (editors G Fischer, U Rehmann), Deutsche Mathematiker Vereinigung (1998) 327

[13] J B Etnyre, L L Ng, J M Sabloff, Invariants of Legendrian knots and coherent orientations, J. Symplectic Geom. 1 (2002) 321

[14] K Fukaya, Application of Floer homology of Langrangian submanifolds to symplectic topology, from: "Morse theoretic methods in nonlinear analysis and in symplectic topology", NATO Sci. Ser. II Math. Phys. Chem. 217, Springer (2006) 231 | DOI

[15] K Irie, A chain level Batalin–Vilkovisky structure in string topology via de Rham chains, Int. Math. Res. Not. 2018 (2018) 4602 | DOI

[16] K Irie, Chain level loop bracket and pseudo-holomorphic disks, J. Topol. 13 (2020) 870 | DOI

[17] J Milnor, Morse theory, 51, Princeton Univ. Press (1963)

[18] L Ng, Knot and braid invariants from contact homology, II, Geom. Topol. 9 (2005) 1603 | DOI

[19] L Ng, Framed knot contact homology, Duke Math. J. 141 (2008) 365 | DOI

[20] C Viterbo, Generating functions, symplectic geometry, and applications, from: "Proceedings of the International Congress of Mathematicians, I" (editor S D Chatterji), Birkhäuser (1995) 537

[21] C A Weibel, An introduction to homological algebra, 38, Cambridge Univ. Press (1994) | DOI