Algebraic & Geometric Topology

On the cohomology algebra of a fiber

Luc Menichi

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Abstract

Let f:EB be a fibration of fiber F. Eilenberg and Moore have proved that there is a natural isomorphism of vector spaces H(F;Fp)TorC(B)(C(E),Fp). Generalizing the rational case proved by Sullivan, Anick [Hopf algebras up to homotopy, J. Amer. Math. Soc. 2 (1989) 417–453] proved that if X is a finite r–connected CW–complex of dimension rp then the algebra of singular cochains C(X;Fp) can be replaced by a commutative differential graded algebra A(X) with the same cohomology. Therefore if we suppose that f:EB is an inclusion of finite r–connected CW–complexes of dimension rp, we obtain an isomorphism of vector spaces between the algebra H(F;Fp) and TorA(B)(A(E),Fp) which has also a natural structure of algebra. Extending the rational case proved by Grivel–Thomas–Halperin [P P Grivel, Formes differentielles et suites spectrales, Ann. Inst. Fourier 29 (1979) 17–37] and [S Halperin, Lectures on minimal models, Soc. Math. France 9-10 (1983)], we prove that this isomorphism is in fact an isomorphism of algebras. In particular, H(F;Fp) is a divided powers algebra and pth powers vanish in the reduced cohomology H̃(F;Fp).

Article information

Source
Algebr. Geom. Topol., Volume 1, Number 2 (2001), 719-742.

Dates
Received: 17 October 2000
Revised: 12 October 2001
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882646

Digital Object Identifier
doi:10.2140/agt.2001.1.719

Mathematical Reviews number (MathSciNet)
MR1875615

Zentralblatt MATH identifier
0981.55006

Subjects
Primary: 55R20: Spectral sequences and homology of fiber spaces [See also 55Txx] 55P62: Rational homotopy theory
Secondary: 18G15: Ext and Tor, generalizations, Künneth formula [See also 55U25] 57T30: Bar and cobar constructions [See also 18G55, 55Uxx] 57T05: Hopf algebras [See also 16T05]

Keywords
homotopy fiber bar construction Hopf algebra up to homotopy loop space homology divided powers algebra

Citation

Menichi, Luc. On the cohomology algebra of a fiber. Algebr. Geom. Topol. 1 (2001), no. 2, 719--742. doi:10.2140/agt.2001.1.719. https://projecteuclid.org/euclid.agt/1513882646


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References

  • C. Allday and V. Puppe, Cohomological methods in transformation groups, Cambridge Univ. Press, 1993.
  • D. J. Anick, Hopf algebras up to homotopy, J. Amer. Math. Soc. 2 (1989), no. 3, 417–453.
  • L. Avramov and S. Halperin, Through the looking glass: a dictionnary between rational homotopy theory and local algebra, Algebra, Algebraic Topology and their Interactions (Stockholm, 1983), Lecture Notes in Math., vol. 1183, Springer, Berlin-New York, 1986, pp. 3–27.
  • H. J. Baues, Algebraic homotopy, Cambridge Univ. Press, 1989.
  • H. J. Baues and J.-M. Lemaire, Minimal models in homotopy theory, Math. Ann. 225 (1977), 219–242.
  • K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics, no. 87, Springer-Verlag, New York, 1994, Corrected reprint of the 1982 original.
  • H. Cartan and Eilenberg S., Homological algebra, Princeton University Press, 1956.
  • N. Dupont and K. Hess, Twisted tensor models for fibrations, J. Pure Appl. Algebra 91 (1994), 109–120.
  • S. Eilenberg and Moore J., Homology and fibrations. I. coalgebras, cotensor product and its derived functors, Comment. Math. Helv. 40 (1966), 199–236.
  • Y. Félix, S. Halperin, and J.-C. Thomas, Adam's cobar equivalence, Trans. Amer. Math. Soc. 329 (1992), 531–549.
  • ––––, Differential graded algebras in topology, Handbook of Algebraic Topology (I. M. James, ed.), North-Holland, Amsterdam, 1995, pp. 829–865.
  • ––––, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, 2000.
  • P. P. Grivel, Formes différentielles et suites spectrales, Ann. Inst. Fourier 29 (1979), 17–37.
  • S. Halperin, Notes on divided powers algebras, written in Sweden.
  • ––––, Lectures on minimal models, Soc. Math. France 9-10 (1983).
  • ––––, Universal enveloping algebras and loop space homology, J. Pure Appl. Algebra 83 (1992), 237–282.
  • D. Husemoller, Fibre bundles, third ed., Graduate Texts in Mathematics, no. 20, Springer-Verlag, New York, 1994.
  • Gugenheim V. K. A. M. and May J. P., On the theory and applications of differential torsion products, Mem. Amer. Math. Soc. 142 (1974).
  • S. Mac Lane, Homology, Springer-Verlag, Berlin, 1963.
  • M. Majewski, Rational homotopical models and uniqueness, Mem. Amer. Math. Soc. 682 (2000).
  • C. A. McGibbon and C. W. Wilkerson, Loop spaces of finite complexes at large primes, Proc. Amer. Math. Soc. 96 (1986), 698–702.
  • Bitjong Ndombol, Algèbres de cochaî nes quasi-commutatives et fibrations algébriques, J. Pure Appl. Algebra 125 (1998), no. 1-3, 261–276.
  • D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331.