## Algebraic & Geometric Topology

### 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 $H∗(F;Fp)≅TorC∗(B)(C∗(E),Fp)$. 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 $f:E→B$ 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 $p$th powers vanish in the reduced cohomology $H̃∗(F;Fp)$.

#### Article information

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

Dates
Revised: 12 October 2001
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882646

Digital Object Identifier
doi:10.2140/agt.2001.1.719

Mathematical Reviews number (MathSciNet)
MR1875615

Zentralblatt MATH identifier
0981.55006

#### Citation

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. https://projecteuclid.org/euclid.agt/1513882646

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