Algebraic & Geometric Topology

On the cohomology algebra of a fiber

Luc Menichi

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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

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

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

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Zentralblatt MATH identifier

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]

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


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.

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