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Sur l’opération d’holonomie rationnelle. (French) Zbl 0594.55011
Algebra, algebraic topology and their interactions, Proc. Conf., Stockholm 1983, Lect. Notes Math. 1183, 136-169 (1986).
[For the entire collection see Zbl 0577.00005.]
The authors consider fibrations $$F\hookrightarrow E\to^{p}B$$, and their discussion of various (rational) homological properties centers around the rational holonomy operation, $$H_*(\Omega B)\otimes H_*(F)\to H_*(F)$$, induced by the holonomy. Both results and techniques illustrate the fruitful interactions between rational homotopy and local algebra, see also the expository paper of L. Avramov and S. Halperin [ibid. 1183, 1-27 (1986; Zbl 0588.13010)]. To quote (a few) results: If dim $$\pi$$ $${}_*(\Omega B)\otimes {\mathbb{Q}}<\infty$$ and dim $$H_*(E)<\infty$$, then $$H_*(F)$$ is a Noetherian $$H_*(\Omega B)$$- module; if B is a suspension, then dim $$H_*(E)<\infty$$ implies that $$H_*(F)$$ is finitely generated as an $$H_*(\Omega B)$$-module. If p is formalisable, one has an algebra isomorphism $$H_*(\Omega B; {\mathbb{Q}})\sim Ext_{H^*(B)}({\mathbb{Q}}, {\mathbb{Q}})$$ and a compatible module isomorphism $$H_*(F; {\mathbb{Q}})\sim Ext_{H^*(B)}(H^*(E), {\mathbb{Q}})$$. There is a spectral sequence going from $$Tor^{H_*(\Omega B)}({\mathbb{Q}},H_*(F))$$ to $$H_*(E)$$, which collapses when B is a suspension....
 55P62 Rational homotopy theory 55T20 Eilenberg-Moore spectral sequences 55R05 Fiber spaces in algebraic topology 55P35 Loop spaces 57T25 Homology and cohomology of $$H$$-spaces