Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 1, Number 2 (2001), 719-742.
On the cohomology algebra of a fiber
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 .
Algebr. Geom. Topol., Volume 1, Number 2 (2001), 719-742.
Received: 17 October 2000
Revised: 12 October 2001
First available in Project Euclid: 21 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 55R20: Spectral sequences and homology of fiber spaces [See also 55Txx] 55P62: Rational homotopy theory
Secondary: 18G15: Ext and Tor, generalizations, Künneth formula [See also 55U25] 57T30: Bar and cobar constructions [See also 18G55, 55Uxx] 57T05: Hopf algebras [See also 16T05]
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