Algebraic and Geometric Topology 1 (2001), paper no. 36, pages 719-742.

On the cohomology algebra of a fiber

Luc Menichi

Abstract. Let f:E-->B be a fibration of fiber F. Eilenberg and Moore have proved that there is a natural isomorphism of vector spaces between H^*(F;F_p) and Tor^{C^*(B)}(C^*(E),F_p). 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+1 then the algebra of singular cochains C^*(X;F_p) can be replaced by a commutative differential graded algebra A(X) with the same cohomology. Therefore if we suppose that f:E-->B is an inclusion of finite r-connected CW-complexes of dimension < rp+1, we obtain an isomorphism of vector spaces between the algebra H^*(F;F_p) and Tor^{A(B)}(A(E),F_p) which has also a natural structure of algebra. Extending the rational case proved by Grivel-Thomas-Halperin [PP 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;F_p) is a divided powers algebra and p-th powers vanish in the reduced cohomology \tilde(H)^*(F;F_p).

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

AMS subject classification. Primary: 55R20, 55P62. Secondary: 18G15, 57T30, 57T05.

DOI: 10.2140/agt.2001.1.719

E-print: arXiv:math.AT/0201134

Submitted: 17 October 2000. (Revised: 12 October 2001.) Accepted: 26 Novemver 2001. Published: 1 December 2001.

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Luc Menichi
Universite d'Angers, Faculte des Sciences
2 Boulevard Lavoisier, 49045 Angers, FRANCE

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