On the cohomology algebra of a fiber

Abstract

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 $H^{\ast }\left(F;{\mathbb{F}}_p\right)\cong {Tor}^{C^{\ast }\left(B\right)}\left(C^{\ast }\left(E\right),{\mathbb{F}}_p\right)$. 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 $\le rp$ then the algebra of singular cochains $C^{\ast }\left(X;{\mathbb{F}}_p\right)$ can be replaced by a commutative differential graded algebra $A\left(X\right)$ with the same cohomology. Therefore if we suppose that $f:E\to B$ is an inclusion of finite $r$–connected CW–complexes of dimension $\le rp$, we obtain an isomorphism of vector spaces between the algebra $H^{\ast }\left(F;{\mathbb{F}}_p\right)$ and ${Tor}^{A\left(B\right)}\left(A\left(E\right),{\mathbb{F}}_p\right)$ 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, $H^{\ast }\left(F;{\mathbb{F}}_p\right)$ is a divided powers algebra and $p$th powers vanish in the reduced cohomology ${\tilde{H}}^{\ast }\left(F;{\mathbb{F}}_p\right)$.