×

Invariants of the Lusternik-Schnirelmann type and the topology of critical sets. (English) Zbl 0618.55003

The authors propose the following generalization of the Lyusternik- Schnirelman category of a space X: If \({\mathcal A}\) is any nontrivial class of spaces \({\mathcal A}\)-cat X is the smallest number k such that there exists a covering \(\{X_ 1,...,X_ k\}\) of X (of a certain kind) for which each inclusion \(X_ j\subset X\) factors through some \(A\in {\mathcal A}\), up to homotopy. If \({\mathcal A}\) consists only of the one point space, \({\mathcal A}\)-cat X\(=cat X\) is the classical L-S category of the space X and in general \({\mathcal A}\)-cat \(X\leq cat X.\)
They present a systematic theory for \({\mathcal A}\)-cat. In particular, they obtain improvements of results of T. Ganea [Ill. J. Math. 11, 417- 427 (1967; Zbl 0149.407)] and some new information on the topological structure of the critical set of a differential real function on a paracompact \(C^ 1\)-Banach manifold satisfying the generalized Palais- Smale condition.
Reviewer: J.C.Thomas

MSC:

55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Citations:

Zbl 0149.407
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] I. Berstein and P. J. Hilton, Category and generalized Hopf invariants, Illinois J. Math. 4 (1960), 437 – 451. · Zbl 0113.38301
[2] I. Berstein and P. J. Hilton, On suspensions and comultiplications, Topology 2 (1963), 73 – 82. · Zbl 0115.40403 · doi:10.1016/0040-9383(63)90024-7
[3] M. Clapp and D. Puppe, The generalized Lusternik-Schnirelmann category of a product space (in preparation). · Zbl 0709.55001
[4] Tammo tom Dieck, Partitions of unity in homotopy theory, Composito Math. 23 (1971), 159 – 167. · Zbl 0212.55804
[5] Tammo tom Dieck, Klaus Heiner Kamps, and Dieter Puppe, Homotopietheorie, Lecture Notes in Mathematics, Vol. 157, Springer-Verlag, Berlin-New York, 1970. · Zbl 0203.25401
[6] E. Fadell, The equivariant Ljusternik-Schnirelmann method for invariant functionals and relative cohomological index theory, Preprint. · Zbl 0575.58014
[7] Ralph H. Fox, On the Lusternik-Schnirelmann category, Ann. of Math. (2) 42 (1941), 333 – 370. · Zbl 0027.43104 · doi:10.2307/1968905
[8] T. Ganea, A generalization of the homology and homotopy suspension, Comment. Math. Helv. 39 (1965), 295 – 322. · Zbl 0142.40702 · doi:10.1007/BF02566956
[9] T. Ganea, Lusternik-Schnirelmann category and strong category, Illinois J. Math. 11 (1967), 417 – 427. · Zbl 0149.40703
[10] W. J. Gilbert, Some examples for weak category and conilpotency, Illinois J. Math. 12 (1968), 421 – 432. · Zbl 0157.54203
[11] Brayton Gray, Homotopy theory, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. An introduction to algebraic topology; Pure and Applied Mathematics, Vol. 64. · Zbl 0322.55001
[12] Morris W. Hirsch, Differential topology, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. · Zbl 0356.57001
[13] I. M. James, On category, in the sense of Lusternik-Schnirelmann, Topology 17 (1978), no. 4, 331 – 348. · Zbl 0408.55008 · doi:10.1016/0040-9383(78)90002-2
[14] L. Lusternik and L. Schnirelmann, Méthodes topologiques dans les problèmes variationels, Hermann, Paris, 1934. · Zbl 0011.02803
[15] Michael Mather, Pull-backs in homotopy theory, Canad. J. Math. 28 (1976), no. 2, 225 – 263. · Zbl 0351.55005 · doi:10.4153/CJM-1976-029-0
[16] Luis Montejano, A quick proof of Singhof’s \?\?\?(\?\times \?\textonesuperior )=\?\?\?(\?)+1 theorem, Manuscripta Math. 42 (1983), no. 1, 49 – 52. · Zbl 0639.55001 · doi:10.1007/BF01171745
[17] -, Lusternik-Schnirelmann category: A geometric approach, Memoirs of the Topology Semester held at the Banach Center, Warsaw 1984 (to appear).
[18] Mitutaka Murayama, On \?-ANRs and their \?-homotopy types, Osaka J. Math. 20 (1983), no. 3, 479 – 512. · Zbl 0531.57034
[19] Richard S. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963), 299 – 340. · Zbl 0122.10702 · doi:10.1016/0040-9383(63)90013-2
[20] Richard S. Palais, Homotopy theory of infinite dimensional manifolds, Topology 5 (1966), 1 – 16. · Zbl 0138.18302 · doi:10.1016/0040-9383(66)90002-4
[21] Richard S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115 – 132. · Zbl 0143.35203 · doi:10.1016/0040-9383(66)90013-9
[22] R. S. Palais and S. Smale, A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1964), 165 – 172. · Zbl 0119.09201
[23] Wilhelm Singhof, Minimal coverings of manifolds with balls, Manuscripta Math. 29 (1979), no. 2-4, 385 – 415. · Zbl 0415.55001 · doi:10.1007/BF01303636
[24] S. Smale, Morse theory and a non-linear generalization of the Dirichlet problem, Ann. of Math. (2) 80 (1964), 382 – 396. · Zbl 0131.32305 · doi:10.2307/1970398
[25] Floris Takens, The Lusternik-Schnirelman categories of a product space, Compositio Math. 22 (1970), 175 – 180. · Zbl 0198.28302
[26] Stefan Waner, Equivariant homotopy theory and Milnor’s theorem, Trans. Amer. Math. Soc. 258 (1980), no. 2, 351 – 368. , https://doi.org/10.1090/S0002-9947-1980-0558178-7 Stefan Waner, Equivariant fibrations and transfer, Trans. Amer. Math. Soc. 258 (1980), no. 2, 369 – 384. , https://doi.org/10.1090/S0002-9947-1980-0558179-9 Stefan Waner, Equivariant classifying spaces and fibrations, Trans. Amer. Math. Soc. 258 (1980), no. 2, 385 – 405. · Zbl 0444.55010
[27] George W. Whitehead, The homology suspension, Colloque de topologie algébrique, Louvain, 1956, Georges Thone, Liège; Masson & Cie, Paris, 1957, pp. 89 – 95. · Zbl 0079.39101
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.