Volume 1, issue 2 (2001)

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On the cohomology algebra of a fiber

Luc Menichi

Algebraic & Geometric Topology 1 (2001) 719–742
 arXiv: math.AT/0201134
Abstract

Let $f:E\to B$ be a fibration of fiber $F$. Eilenberg and Moore have proved that there is a natural isomorphism of vector spaces ${H}^{\ast }\left(F;{\mathbb{F}}_{p}\right)\cong {Tor}^{{C}^{\ast }\left(B\right)}\left({C}^{\ast }\left(E\right),{\mathbb{F}}_{p}\right)$. 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 $\le rp$ then the algebra of singular cochains ${C}^{\ast }\left(X;{\mathbb{F}}_{p}\right)$ can be replaced by a commutative differential graded algebra $A\left(X\right)$ with the same cohomology. Therefore if we suppose that $f:E\to B$ is an inclusion of finite $r$–connected CW–complexes of dimension $\le rp$, we obtain an isomorphism of vector spaces between the algebra ${H}^{\ast }\left(F;{\mathbb{F}}_{p}\right)$ and ${Tor}^{A\left(B\right)}\left(A\left(E\right),{\mathbb{F}}_{p}\right)$ 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, ${H}^{\ast }\left(F;{\mathbb{F}}_{p}\right)$ is a divided powers algebra and $p$th powers vanish in the reduced cohomology ${\stackrel{̃}{H}}^{\ast }\left(F;{\mathbb{F}}_{p}\right)$.

Keywords
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