Let be a fibration of fiber . Eilenberg and Moore have proved that there is a natural isomorphism of vector spaces . Generalizing the rational case proved by Sullivan, Anick [Hopf algebras up to homotopy, J. Amer. Math. Soc. 2 (1989) 417–453] proved that if is a finite –connected CW–complex of dimension then the algebra of singular cochains can be replaced by a commutative differential graded algebra with the same cohomology. Therefore if we suppose that is an inclusion of finite –connected CW–complexes of dimension , we obtain an isomorphism of vector spaces between the algebra and 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, is a divided powers algebra and th powers vanish in the reduced cohomology .
"On the cohomology algebra of a fiber." Algebr. Geom. Topol. 1 (2) 719 - 742, 2001. https://doi.org/10.2140/agt.2001.1.719