Mathematics – Algebraic Topology
Scientific paper
2002-01-15
Algebr. Geom. Topol. 1 (2001) 719-742
Mathematics
Algebraic Topology
Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-36.abs.html
Scientific paper
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 \mathaccent "707E {H}^*(F;F_p).
No associations
LandOfFree
On the cohomology algebra of a fiber does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On the cohomology algebra of a fiber, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the cohomology algebra of a fiber will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-554145