On the cohomology algebra of a fiber

Let $f : E \to B$ be a fibration of fiber $F$ . Eilenberg and Moore have proved that there is a natural isomorphism of vector spaces $H^{*} (F; F_{p}) ≅ {Tor}^{C^{*} (B)} (C^{*} (E), F_{p})$ . 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 $\leq r p$ 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 $f : E \to B$ is an inclusion of finite $r$ –connected CW–complexes of dimension $\leq r p$ , 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 [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, $H^{*} (F; F_{p})$ 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

Mathematical Subject Classification 2000

Primary: 55R20, 55P62

Secondary: 18G15, 57T30, 57T05

References

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Publication

Received: 17 October 2000

Revised: 12 October 2001

Accepted: 26 2001

Published: 1 December 2001

Authors

Luc Menichi
Université d’Angers Faculté des Sciences 2 Boulevard Lavoisier 49045 Angers FRANCE

Volume 1, issue 2 (2001)

Luc Menichi