Algebraic & Geometric Topology

Splitting of Gysin extensions

A J Berrick and A A Davydov

Full-text: Open access


Let XB be an orientable sphere bundle. Its Gysin sequence exhibits H(X) as an extension of H(B)–modules. We prove that the class of this extension is the image of a canonical class that we define in the Hochschild 3–cohomology of H(B), corresponding to a component of its A–structure, and generalizing the Massey triple product. We identify two cases where this class vanishes, so that the Gysin extension is split. The first, with rational coefficients, is that where B is a formal space; the second, with integer coefficients, is where B is a torus.

Article information

Algebr. Geom. Topol., Volume 1, Number 2 (2001), 743-762.

Received: 11 October 2000
Revised: 17 July 2001
First available in Project Euclid: 21 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16E45: Differential graded algebras and applications 55R25: Sphere bundles and vector bundles 55S35: Obstruction theory
Secondary: 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.) 55R20: Spectral sequences and homology of fiber spaces [See also 55Txx] 55S20: Secondary and higher cohomology operations 55S30: Massey products

Gysin sequence Hochschild homology differential graded algebra formal space $A_{\infty}$–structure Massey triple product


Berrick, A J; Davydov, A A. Splitting of Gysin extensions. Algebr. Geom. Topol. 1 (2001), no. 2, 743--762. doi:10.2140/agt.2001.1.743.

Export citation


  • H. Cartan, S. Eilenberg, Homological Algebra, Princeton Univ. Press (Princeton NJ, 1956).
  • P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245–274.
  • N. Dupont, M. Vigu-Poirrier, Formalité des espaces de lacets libres, Bull. Soc. Math. France 126 (1998), 141–148.
  • T. Ekedahl, Two examples of smooth projective varieties with nonzero Massey products, Algebra, Algebraic Topology and their Interactions (Stockholm, 1983), Lecture Notes in Math 1183, Springer (Berlin, 1986), 128–132.
  • E. Farjoun, Cellular Spaces, Null Spaces and Homotopy Localization, Lect. Notes in Math. 1622, Springer (Berlin, 1996).
  • M. Gerstenhaber, The cohomology structure of an associative ring, Annals of Math. 78 (1963), 267-288.
  • G. Hirsch, L'anneau de cohomologie d'un espace fibré en sphères, C. R. Acad. Sci. Paris 241 (1955), 1021-1023.
  • T. V. Kadeishvili, On the theory of homology groups of fibre spaces, Uspehi Mat. Nauk. 35:3 (1980), 183-188; English translation in Russian Math Surveys 35:3 (1980).
  • T. V. Kadeishvili, The structure of the $A(\infty)$-algebra, and the Hochschild and Harrison cohomologies, (Russian; English summary) Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk. Gruzin S.S.R. 91 (1988), 19–27.
  • D. Kraines, Massey higher products, Trans. Amer. Math. Soc. 124 (1966), 431-449.
  • S. Kumar, Rational homotopy theory of flag varieties associated to Kac-Moody groups, Infinite-dimensional Groups with Applications (Berkeley, Calif, 1984), Math. Sci. Res. Inst. Publ. 4, Springer (New York, 1985), 233–273.
  • S-T. Lee, J. Packer, The cohomology of the integer Heisenberg group, J. Algebra 184 (1996), 230-250.
  • D. G. Malm, Concerning the cohomology ring of a sphere bundle, Pacific J. Math. 9 (1959), 1191-1214.
  • W. S. Massey, On the cohomology ring of a sphere bundle, J. Math. Mech. 7 (1958), 265-289.
  • T. J. Miller, On the formality of $(k-1)$-connected compact manifolds of dimension less than or equal to $4k-2$, Illinois J. Math. 23 (1979), 253–258.
  • J. W. Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 48 (1978), 137–204.
  • M. Parhizgar, On the cohomology ring of the free loop space of a wedge of spheres, Math. Scand. 80 (1997), 195–248.
  • J. J. Rotman, An Introduction to Algebraic Topology, Graduate Texts in Math. 119, Springer (New York, 1988).
  • E. H. Spanier, Algebraic Topology, McGraw-Hill (New York, 1966).
  • J. D. Stasheff, Homotopy associativity of H-spaces. II, Trans. Amer. Math. Soc. 108 (1963), 293-312.