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Sur l’opĂ©ration 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....
Reviewer: St.Papadima

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