Rational approximations to L-S category and a conjecture of Ganea
Author:
Barry Jessup
Journal:
Trans. Amer. Math. Soc. 317 (1990), 655-660
MSC:
Primary 55P62; Secondary 55P50
DOI:
https://doi.org/10.1090/S0002-9947-1990-0956033-8
MathSciNet review:
956033
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Abstract | References | Similar Articles | Additional Information
Abstract: The rational version of Ganea's conjecture for L-S category, namely that , if
is a rational space and
denotes the
-sphere, is still open. Recently, a module type approximation to
, was introduced by Halperin and Lemaire. We have previously shown that
satisfies Ganea's conjecture. Here we show that for
connected
, if
is at least
, then
. This yields Ganea's conjecture for these spaces. We also extend other properties of
, previously unknown for cat, to these spaces.
- [Av-Ha] Luchezar Avramov and Stephen Halperin, Through the looking glass: a dictionary between rational homotopy theory and local algebra, Algebra, algebraic topology and their interactions (Stockholm, 1983) Lecture Notes in Math., vol. 1183, Springer, Berlin, 1986, pp. 1–27. MR 846435, https://doi.org/10.1007/BFb0075446
- [Fe-Ha] Yves Félix and Stephen Halperin, Rational LS category and its applications, Trans. Amer. Math. Soc. 273 (1982), no. 1, 1–38. MR 664027, https://doi.org/10.1090/S0002-9947-1982-0664027-0
- [Fo] Ralph H. Fox, On the Lusternik-Schnirelmann category, Ann. of Math. (2) 42 (1941), 333–370. MR 4108, https://doi.org/10.2307/1968905
- [Ga
] Tudor Ganea, Lusternik-Schnirelmann category and cocategory, Proc. London Math. Soc. (3) 10 (1960), 623–639. MR 0126278, https://doi.org/10.1112/plms/s3-10.1.623
- [Ga
] Tudor Ganea, Some problems on numerical homotopy invariants, Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle Wash., 1971) Springer, Berlin, 1971, pp. 23–30. Lecture Notes in Math., Vol. 249. MR 0339147
- [Ha-Le] S. Halperin and J.-M. Lemaire, Notions of category in differential algebra, Algebraic topology—rational homotopy (Louvain-la-Neuve, 1986) Lecture Notes in Math., vol. 1318, Springer, Berlin, 1988, pp. 138–154. MR 952577, https://doi.org/10.1007/BFb0077800
- [Ha] S. Halperin, Lectures on minimal models, Mém. Soc. Math. France (N.S.) 9-10 (1983), 261. MR 736299
- [J] B. Jessup, Rational Lusternik-Schnirelmann category, fibrations and a conjecture of Ganea, Preprint.
- [Ja] I. M. James, On category, in the sense of Lusternik-Schnirelmann, Topology 17 (1978), no. 4, 331–348. MR 516214, https://doi.org/10.1016/0040-9383(78)90002-2
- [Lu-Sc] L. Lusternik and L. Schnirelmann, Méthodes topologiques dans les problèmes variationnels, Hermann, Paris, 1934.
- [Si] Wilhelm Singhof, Minimal coverings of manifolds with balls, Manuscripta Math. 29 (1979), no. 2-4, 385–415. MR 545050, https://doi.org/10.1007/BF01303636
- [Su] Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331 (1978). MR 646078
- [To] Graham Hilton Toomer, Lusternik-Schnirelmann category and the Moore spectral sequence, Math. Z. 138 (1974), 123–143. MR 356037, https://doi.org/10.1007/BF01214229
- [Wh] George W. Whitehead, The homology suspension, Colloque de topologie algébrique, Louvain, 1956, Georges Thone, Liège; Masson & Cie, Paris, 1957, pp. 89–95. MR 0094794
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1990-0956033-8
Keywords:
Lusternik-Schnirelmann category,
minimal models,
rational homotopy
Article copyright:
© Copyright 1990
American Mathematical Society