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On the cohomology algebra of a fiber. (English) Zbl 0981.55006

For a fibration \(f:E\rightarrow B\) of fiber \(F\) and simple connected base \(B\) , the well-known Eilenberg-Moore isomorphism \(H^{*}(F;{\mathbf k})\cong Tor^{C^{*}(B)}(C^{*}(E);{\mathbf k})\) is an isomorphism of graded vector spaces and in general it does not give the multiplication structure of \(H^{*}(F;{\mathbf k})\). However, in the paper under review it is proved that for \(\operatorname {char}{\mathbf k}\) sufficiently large and a natural multiplication structure on \(Tor\), the Eilenberg-Moore isomorphism becomes one of graded algebras. The algebra of singular cochains \(C^{*}(X)\) is naturally linked to a commutative differential graded algebra \(A(X)\), inducing isomorphism in homology. The main results of the paper are the following two theorems:
Theorem A. Assume the characteristic of the field \({\mathbf k}\) is an odd prime \(p\) and consider an inclusion \(F\hookrightarrow E\) of finite \(r\)-connected CW-complexes \((r\geq 1\)) of dimension \(\leq rp\). Then the Eilenberg isomorphism \(H^{*}(F;{\mathbf k})\cong \text{Tor}^{A(B)}(A(E);{\mathbf k})\) is an isomorphism of graded algebra.
Theorem B. Let \(p\) be an odd prime. Consider the homotopy fiber \(F\) of an inclusion of finite \(r\)-connected CW-complexes of dimension \(\leq rp\). Then the cohomology algebra \(H^{*}(F;F_{p})\) is a divided powers algebra. In particular, \(p\)th powers vanish in the reduced cohomology \(\widetilde{H}^{*}(F;F_{p})\).

MSC:

55R20 Spectral sequences and homology of fiber spaces in algebraic topology
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
55P62 Rational homotopy theory
57T30 Bar and cobar constructions
57T05 Hopf algebras (aspects of homology and homotopy of topological groups)
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