×

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

MSC:

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
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.