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On the cohomology algebra of a fiber

Luc Menichi

Algebraic & Geometric Topology 1 (2001) 719–742

arXiv: math.AT/0201134


Let f : EB 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 : EB 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 pth powers vanish in the reduced cohomology H̃(F; Fp).

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
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Received: 17 October 2000
Revised: 12 October 2001
Accepted: 26 2001
Published: 1 December 2001
Luc Menichi
Université d’Angers
Faculté des Sciences
2 Boulevard Lavoisier
49045 Angers