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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