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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[1] ANDREWS (P.) et ARKOWITZ (M.) . - Sullivan’s minimal models and higher order Whitehead products , Canad. J. Math., vol. XXX, n^\circ 5, 1978 , p. 961-982. MR 80b:55008 | Zbl 0441.55012 · Zbl 0441.55012
[2] BOUSFIELD (A. K.) et GUGENHEIM (V. K. A. M.) . - On the P. L. de Rham theory and rational homotopy type , Memoirs of A.M.S., 179, 1976 . Zbl 0338.55008 · Zbl 0338.55008
[3] BAUES (H. J.) et LEMAIRE (J. M.) . - Minimal models in homotopy theory , Math. Ann., vol. 225, 1977 , p. 219-242. MR 55 #4174 | Zbl 0322.55019 · Zbl 0322.55019
[4] FELIX (Y.) et HALPERIN (S.) . - Rational L. S. category and its applications (à paraître Trans. of A.M.S.). Zbl 0508.55004 · Zbl 0508.55004
[5] FÉLIX (Y.) , HALPERIN (S.) et THOMAS (J. C.) . - The homotopy Lie algebra for finite complexes (à paraître).
[6] GINSBURG (M.) . - On the L. S. category , Ann. of Math., vol. 77, 1963 , p. 538-551. MR 26 #6976 | Zbl 0148.17003 · Zbl 0148.17003
[7] LEMAIRE (J. M.) et SIGRIST (F.) . - Sur les invariants d’homotopie rationnelle liés à la L. S. catégorie , Comment Math. Helvetici, vol. 56, 1981 , p. 103-122. MR 82g:55009 | Zbl 0479.55008 · Zbl 0479.55008
[8] SULLIVAN (D.) . - Infinitesimal computations in topology , Publ. I.H.E.S., vol. 47, 1977 , p. 269-331. Numdam | MR 58 #31119 | Zbl 0374.57002 · Zbl 0374.57002
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