#### 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.

Notes on file formats
Luc Menichi

Universite d'Angers, Faculte des Sciences

2 Boulevard Lavoisier, 49045 Angers, FRANCE

Email: Luc.Menichi@univ-angers.fr

AGT home page

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