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2001 On the cohomology algebra of a fiber
Luc Menichi
Algebr. Geom. Topol. 1(2): 719-742 (2001). DOI: 10.2140/agt.2001.1.719


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).


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Luc Menichi. "On the cohomology algebra of a fiber." Algebr. Geom. Topol. 1 (2) 719 - 742, 2001.


Received: 17 October 2000; Revised: 12 October 2001; Published: 2001
First available in Project Euclid: 21 December 2017

zbMATH: 0981.55006
MathSciNet: MR1875615
Digital Object Identifier: 10.2140/agt.2001.1.719

Primary: 55P62, 55R20
Secondary: 18G15, 57T05, 57T30

Rights: Copyright © 2001 Mathematical Sciences Publishers


Vol.1 • No. 2 • 2001
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