Hodge Filtration and Operations in Higher Hochschild (Co)homology and Applications to Higher String Topology

  • Grégory GinotEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2194)


This paper is based on lectures given at the Vietnamese Institute for Advanced Studies in Mathematics and aims to present the theory of higher Hochschild (co)homology and its application to higher string topology. There is an emphasis on explicit combinatorial models provided by simplicial sets to describe derived structures carried or described by Higher Hochschild (co)homology functors. It contains detailed proofs of results stated in a previous note as well as some new results. One of the main result is a proof that string topology for higher spheres inherits a Hodge filtration compatible with an (homotopy) E n+1-algebra structure on the chains for d-connected Poincaré duality spaces. We also prove that the E n -centralizer of maps of commutative (dg-)algebras are equipped with a Hodge decomposition and a compatible structure of framed E n -algebras. We also study Hodge decompositions for suspensions and products by spheres, both as derived functors and combinatorially.


  1. [AF]
    D. Ayala, J. Francis, Factorization homology of topological manifolds. J. Topol. 8(4), 1045–1084 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [ArTu]
    G. Arone, V. Turchin, Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots. Ann. Inst. Fourier (Grenoble) 65(1), 1–62 (2015)Google Scholar
  3. [AT]
    M. Atiyah, D. Tall, Group representations, λ-rings and the J-homomorphism. Topology 8, 253–297 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [BD]
    A. Beilinson, V. Drinfeld, Chiral Algebras. American Mathematical Society Colloquium Publications, vol. 51 (American Mathematical Society, Providence, RI, 2004)Google Scholar
  5. [BHM]
    M. Bökstedt, W.C. Hsiang, I. Madsen, The cyclotomic trace and algebraic K-theory of spaces. Invent. Math. 111(3), 465–539 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [BL]
    J. Block, A. Lazarev, André-Quillen cohomology and rational homotopy of function spaces. Adv. Math. 193(1), 18–39 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [BNT]
    P. Bressler, R. Nest, B. Tsygan, Riemann-Roch theorems via deformation quantization. I, II. Adv. Math. 167(1), 1–25, 26–73 (2002)Google Scholar
  8. [BW]
    N. Bergeron, L. Wolfgang, The decomposition of Hochschild cohomology and Gerstenhaber operations. J. Pure Appl. Algebra 79, 109–129 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [C1]
    K. Costello, Topological conformal field theories and gauge theories. Geom. Topol. 11, 1539–1579 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [C2]
    K. Costello, Topological conformal field theories and Calabi-Yau categories. Adv. Math. 210(1), 165–214 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [Ca]
    A. Căldăraru, The Mukai pairing. II. The Hochschild-Kostant-Rosenberg isomorphism. Adv. Math. 194(1), 34–66 (2005)zbMATHGoogle Scholar
  12. [CaTu]
    A. Căldăraru, J. Tu, Curved A algebras and Landau-Ginzburg models. N. Y. J. Math. 19, 305–342 (2013)MathSciNetzbMATHGoogle Scholar
  13. [CG]
    K. Costello, O. Gwilliam, Factorization algebras in perturbative quantum field theory. Online wiki available at
  14. [Ch]
    K.-T. Chen, Iterated integrals of differential forms and loop space homology. Ann. Math. (2) 97, 217–246 (1973)Google Scholar
  15. [CJ]
    R. Cohen, J. Jones, A homotopy theoretic realization of string topology. Math. Ann. 324(4), 773–798 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [Co]
    F.R. Cohen, The homology of Open image in new window-spaces, n ≥ 0, in The Homology of Iterated Loop Spaces. Lecture Notes in Mathematics, ed. by F.R.Cohen, T.J. Lada, J.P. May, vol. 533 (Springer, Berlin, 1976)Google Scholar
  17. [CPTVV]
    D. Calaque, T. Pantev, B. Toën, M. Vaquié, G. Vezzosi, Shifed Poisson structures. J. Topol. (to appear)Google Scholar
  18. [CS]
    M. Chas, D. Sullivan, String topology. arXiv:math/9911159Google Scholar
  19. [CV]
    R. Cohen, A. Voronov, Notes on string topology, in String Topology and Cyclic Homology. Advance Courses in Mathematics CRM Barcelona (Birkhäuser, Basel, 2006), pp. 1–95Google Scholar
  20. [CW]
    D. Calaque, T. Willwacher, Triviality of the higher formality theorem. Preprint arXiv:1310.4605Google Scholar
  21. [DP]
    V. Dolgushev, B. Paljug, Tamarkin’s construction is equivariant with respect to the action of the Grothendieck-Teichmueller group. J. Homotopy Relat. Struct. 11(3), 503–552 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [Du]
    G. Dunn, Tensor product of operads and iterated loop spaces. J. Pure Appl. Algebra 50(3), 237–258 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [F]
    J. Francis, The tangent complex and Hochschild cohomology of E n-rings. Preprint AT/1104.0181 Google Scholar
  24. [FG]
    J. Francis, D. Gaitsgory, Chiral Koszul duality. Selecta Math. New Ser. 18, 27–87 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [Fr1]
    B. Fresse, Théorie des opérades de Koszul et homologie des algèbres de Poisson. Ann. Math. Blaise Pascal 13(2), 237–312 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [Fr2]
    B. Fresse, Iterated bar complexes of E-infinity algebras and homology theories. Algebr. Geom. Topol. 11(2), 747–838 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. [Fr3]
    B. Fresse, Modules over Operads and Functors. Lecture Notes in Mathematics, vol. 1967 (Springer, Berlin, 2009)Google Scholar
  28. [Fr4]
    B. Fresse, Homotopy of Operads & Grothendieck-Teichmüller Groups. Mathematical Surveys and Monographs, vol. 217 (American Mathematical Society, Providence, 2017)Google Scholar
  29. [FT]
    Y. Félix, J.-C. Thomas, Rational BV-algebra in string topology. Bull. Soc. Math. France 136(2), 311–327 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  30. [FTV]
    Y. Félix, J.-C. Thomas, M. Vigué, The Hochschild cohomology of a closed manifold. Publ. Math. Inst. Hautes Études Sci. 99, 235–252 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  31. [G]
    M. Gerstenhaber, The cohomology structure of an associative ring. Ann. Math. 78(2), 267–288 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  32. [Gi1]
    G. Ginot, Homologie et modèle minimal des algèbres de Gerstenhaber. Ann. Math. Blaise Pascal 11(1), 95–127 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  33. [Gi2]
    G. Ginot, On the Hochschild and Harrison (co)homology of C -algebras and applications to string topology, in Deformation Spaces. Aspects of Mathematics, vol. E40 (Springer, Berlin, 2010), pp. 1–51Google Scholar
  34. [Gi3]
    G. Ginot, Higher order Hochschild cohomology. C. R. Math. Acad. Sci. Paris 346(1–2), 5–10 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  35. [Gi4]
    G. Ginot, Notes on factorization algebras and factorization homology, 124 pp., in Mathematical Aspects of Field Theories. Springer, Mathematical Physics Studies, vol. 5, Part IV (Springer, Cham, 2015), pp. 429–552Google Scholar
  36. [GiRo]
    G. Ginot, M. Robalo, Hochschild-Kostant-Rosenberg Theorem in derived geometry (in preparation)Google Scholar
  37. [GJ]
    P. Goerss, J. Jardine, Simplicial Homotopy Theory. Modern Birkhäuser Classics, 1st edn. (Birkhäuser, Basel, 2009)Google Scholar
  38. [Go]
    T.G. Goodwillie, Cyclic homology, derivations, and the free loopspace. Topology 24(2), 187–215 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  39. [GS]
    M. Gerstenhaber, S. Schack, A Hodge-type decomposition for commutative algebra cohomology. J. Pure Appl. Algebra 48(3), 229–247 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  40. [GTZ]
    G. Ginot, T. Tradler, M. Zeinalian, A Chen model for mapping spaces and the surface product. Ann. Sc. de l’Éc. Norm. Sup., 4e série, t. 43, 811–881 (2010)Google Scholar
  41. [GTZ2]
    G. Ginot, T. Tradler, M. Zeinalian, Derived higher Hochschild homology, topological chiral homology and factorization algebras. Commun. Math. Phys. 326, 635–686 (2014)CrossRefzbMATHGoogle Scholar
  42. [GTZ3]
    G. Ginot, T. Tradler, M. Zeinalian, Higher Hochschild cohomology of E -algebras, Brane topology and centralizers of E n-algebra maps (2015). PreprintGoogle Scholar
  43. [GY]
    G. Ginot, S. Yalin, Deformation theory of bialgebras, higher Hochschild cohomology and formality, avec S. Yalin. Preprint arXiv:1606.01504Google Scholar
  44. [H]
    H. Hiller, λ-rings and algebraic K-theory. J. Pure Appl. Algebra 20(3), 241–266 (1981)Google Scholar
  45. [Hi]
    V. Hinich, Homological algebra of homotopy algebras. Commun. Algebra 25(10), 3291–3323 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  46. [Ho]
    G. Horel, Higher Hochschild cohomology of the Lubin-Tate ring spectrum. Algebr. Geom. Topol. 15(6), 3215–3252 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  47. [K]
    M. Kontsevich, Operads and motives in deformation quantization. Lett. Math. Phys. 48, 35–72 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  48. [Ka]
    D. Kaledin, Motivic structures in non-commutative geometry, in Proceedings of the International Congress of Mathematicians. Volume II, vol. 461–496 (Hindustan Book Agency, New Delhi, 2010)Google Scholar
  49. [KKL]
    L. Katzarkov, M. Kontsevich, T. Pantev, Hodge theoretic aspects of mirror symmetry, in From Hodge Theory to Integrability and TQFT tt*-Geometry. Proceedings of Symposia in Pure Mathematics, vol. 78 (American Mathematical Society, Providence, RI, 2008), pp. 87–174Google Scholar
  50. [Kn]
    B. Knudsen, Higher enveloping algebras. arXiv:1605.01391Google Scholar
  51. [Kr]
    C. Kratzer, λ-structure en K-théorie algébrique. Comment. Math. Helv. 55(2), 233–254 (1980)Google Scholar
  52. [KS]
    M. Kashiwara, P. Schapira, Deformation quantization modules. Astérisque 345, xii+147 pp. (2012)Google Scholar
  53. [KS1]
    M. Kontsevich, Y. Soibelman. Deformation Theory, Volume 1. Unpublished book draft. Available at soibel/
  54. [KS2]
    M. Kontsevich, Y. Soibelman, Notes on A -algebras, A -categories and non-commutative geometry, in Homological Mirror Symmetry. Lecture Notes in Physics, vol. 757 (Springer, Berlin, 2009), pp. 153–219Google Scholar
  55. [L1]
    J.-L. Loday, Opérations sur l’homologie cyclique des algèbres commutatives. Invent. Math. 96(1), 205–230 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  56. [L2]
    J.-L. Loday, Cyclic Homology. Grundlehren der mathematischen Wissenschaften, vol. 301 (Springer, Berlin, 1992)Google Scholar
  57. [LS]
    P. Lambrechts, D. Stanley, Poincaré duality and commutative differential graded algebras. Ann. Sci. Éc. Norm. Supér. (4) 41(4), 495–509 (2008)Google Scholar
  58. [Lu1]
    J. Lurie, Higher Topos Theory. Annals of Mathematics Studies, vol. 170 (Princeton University Press, Princeton, NJ, 2009), xviii+925 pp.Google Scholar
  59. [Lu2]
    J. Lurie, On the classification of topological field theories. Preprint, arXiv:0905.0465v1Google Scholar
  60. [Lu3]
    J. Lurie, Higher Algebra. Book, available at
  61. [MCa]
    R. McCarthy, On operations for Hochschild homology. Commun. Algebra 21(8), 2947–2965 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  62. [P]
    T. Pirashvili, Hodge Decomposition for higher order Hochschild homology. Ann. Sci. École Norm. Sup. (4) 33(2), 151–179 (2000)Google Scholar
  63. [PTVV]
    T. Pantev, B. Toën, M. Vaquié, G. Vezzosi, Shifted symplectic structures. ubl. Math. Inst. Hautes Études Sci. 117, 271–328 (2013)Google Scholar
  64. [R]
    C. Rezk, A model for the homotopy theory of homotopy theory. Trans. Am. Math. Soc. 353(3), 937–1007 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  65. [RZ]
    B. Richter, S. Ziegenhagen, A spectral sequence for the homology of a finite algebraic delooping. J. K-Theory 13(3), 563–599 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  66. [S1]
    D. Sullivan, Infinitesimal computations in topology. Inst. Hautes Études Sci. Publ. Math. 47, 269–331 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  67. [SW]
    P. Salvatore, N. Wahl, Framed discs operads and Batalin Vilkovisky algebras. Q. J. Math. 54(2), 213–231 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  68. [Ta]
    D. Tamarkin, Deformation complex of a d-algebra is a (d+1)-algebra. preprint arXiv:math/0010072Google Scholar
  69. [Tr]
    T. Tradler, The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products. Ann. Inst. Fourier 58(7), 2351–2379 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  70. [TV1]
    B. Toën, G. Vezzosi, Homotopical Algebraic Geometry II: Geometric Stacks and Applications. Memoirs of the American Mathematical Society, vol. 193(902) (American Mathematical Society, Providence, RI, 2008)Google Scholar
  71. [TV2]
    B. Toën, G. Vezzosi, Algèbres simpliciales S 1-équivariantes et théorie de de Rham. Compos. Math. 147(6), 1979–2000 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  72. [TV3]
    B. Toën, G. Vezzosi, A note on Chern character, loop spaces and derived algebraic geometry, in Abel Symposium, Oslo, vol. 4 (2007), pp. 331–354zbMATHGoogle Scholar
  73. [TW]
    V. Turchin, T. Willwacher, Hochschild-Pirashvili homology on suspensions and representations of Out(F n). arXiv:1507.08483Google Scholar
  74. [TZ]
    T. Tradler, M. Zeinalian, Infinity structure of Poincaré duality spaces. Algebr. Geom. Topol. 7, 233–260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  75. [U]
    M. Ungheretti, Free loop space and the cyclic bar construction. arXiv:1602.09035Google Scholar
  76. [VB]
    M. Vigué-Poirrier, D. Burghelea, A model for cyclic homology and algebraic K-theory of 1-connected topological spaces. J. Differ. Geom. 22(2), 243–253 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  77. [W]
    N. Wahl, Universal operations in Hochschild homology. J. Reine Angew. Math. 720, 81–127 (2016)MathSciNetzbMATHGoogle Scholar
  78. [We]
    C. Weibel, An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics, vol. 38 (Cambridge University Press, Cambridge, 1994)Google Scholar
  79. [We2]
    C. Weibel, The Hodge filtration and cyclic homology. K-Theory 12(2), 145–164 (1997)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Laboratoire Analyse, Géométrie et Applications, UMR 7539, Institut Galilée, Université Paris 13VilletaneuseFrance

Personalised recommendations