L. S. catégorie et suite spectrale de Milnor-Moore. (Une nuit dans le train). (French) Zbl 0538.55007

The results in a previous paper by the same authors [Publ. Math., Inst. Hautes Étud. Sci. 56, 179-202 (1982; Zbl 0504.55005)] have created the feeling that the homotopy Lie algebra \(\pi_*(\Omega S)\otimes {\mathbb{Q}}\) rarely happens to be Abelian and that there should be some kind of classification. The main result of the paper under review confirms this: the \({\mathbb{Q}}\)-type of a 1-connected S with Abelian homotopy Lie algebra is uniquely determined by the image of the rational Hurewicz map, which may be arbitrarily prescribed, as soon as the length of the Milnor-Moore filtration on \(H^*(S)\) does not exceed two. It has been previously shown by G. H. Toomer [Math. Z. 138, 123-143 (1974; Zbl 0287.55007)] that this length gives a lower bound for the rational Ljusternik-Schnirelman category of S. Using the main result, the authors exhibit an example of a sequence of 1-connected finite complexes \(S_ k\) with Milnor-Moore filtration length constant and equal to two and satisfying \(cat_ 0(S_ k)>k\), any k. The reference [H] means S. Halperin [Trans. Am. Math. Soc. 230, 173-199 (1977; Zbl 0364.55014)].
Reviewer: S.Papadima


55P62 Rational homotopy theory
55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
55T99 Spectral sequences in algebraic topology
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