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Invariants of the Lusternik-Schnirelmann type and the topology of critical sets


Authors: Mónica Clapp and Dieter Puppe
Journal: Trans. Amer. Math. Soc. 298 (1986), 603-620
MSC: Primary 55M30; Secondary 55P50
DOI: https://doi.org/10.1090/S0002-9947-1986-0860382-0
MathSciNet review: 860382
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Abstract: We introduce and study in detail generalizations of the notion of Lusternik-Schnirelmann category which give information about the topology of the critical set of a differentiable function. We also improve a result of T. Ganea about the equality of the strong category and the category (even in the classical case).


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  • [1] I. Berstein and P. J. Hilton, Category and generalized Hopf invariants, Illinois J. Math. 4 (1960), 437-451. MR 0126276 (23:A3572)
  • [2] -, On suspensions and comultiplications, Topology 2 (1963), 73-82. MR 0150775 (27:762)
  • [3] M. Clapp and D. Puppe, The generalized Lusternik-Schnirelmann category of a product space (in preparation).
  • [4] T. tom Dieck, Partitions of unity in homotopy theory, Compositio Math. 23 (1971), 159-167. MR 0293625 (45:2702)
  • [5] T. tom Dieck, K. H. Kamps and D. Puppe, Homotopietheorie, Lecture Notes in Math., vol. 157, Springer-Verlag, Berlin and New York, 1970. MR 0407833 (53:11603)
  • [6] E. Fadell, The equivariant Ljusternik-Schnirelmann method for invariant functionals and relative cohomological index theory, Preprint.
  • [7] R. H. Fox, On the Lusternik-Schnirelmann category, Ann. of Math. (2) 42 (1941), 333-370. MR 0004108 (2:320f)
  • [8] T. Ganea, A generalization of the homology and homotopy suspension, Comment. Math. Helv. 39 (1965), 295-322. MR 0179791 (31:4033)
  • [9] -, Lusternik-Schnirelmann category and strong category, Illinois J. Math. 11 (1967), 417-427. MR 0229240 (37:4814)
  • [10] W. J. Gilbert, Some examples for weak category and conilpotency, Illinois J. Math. 12 (1968), 421-432. MR 0231375 (37:6930)
  • [11] B. Gray, Homotopy theory. An introduction to algebraic topology, Academic Press, New York, 1975. MR 0402714 (53:6528)
  • [12] M. W. Hirsch, Differential topology, Graduate Texts in Math., no. 33, Springer-Verlag, Berlin and New York, 1976. MR 0448362 (56:6669)
  • [13] I. M. James, On category in the sense of Lusternik-Schnirelmann, Topology 17 (1978), 331-348. MR 516214 (80i:55001)
  • [14] L. Lusternik and L. Schnirelmann, Méthodes topologiques dans les problèmes variationels, Hermann, Paris, 1934.
  • [15] M. Mather, Pull-backs in homotopy theory, Canad. J. Math. 28 (1976), 225-263. MR 0402694 (53:6510)
  • [16] L. Montejano, A quick proof of Singhof's $ \operatorname{cat} (M \times {S^1}) = \operatorname{cat} (M) + 1$ theorem, Manuscripta Math. 42 (1982), 49-52. MR 693418 (85a:55002)
  • [17] -, Lusternik-Schnirelmann category: A geometric approach, Memoirs of the Topology Semester held at the Banach Center, Warsaw 1984 (to appear).
  • [18] M. Murayama, On $ G{\text{ - }}ANR$'s and their $ G$-homotopy types, Osaka J. Math. 20 (1983), 479-512. MR 718960 (85f:57022)
  • [19] R. S. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963), 299-340. MR 0158410 (28:1633)
  • [20] -, Homotopy theory of infinite dimensional manifolds, Topology 5 (1966), 1-16. MR 0189028 (32:6455)
  • [21] -, Lusternik-Schnirelmann theory on Banach manifolds, Topology 5 (1966), 115-132. MR 0259955 (41:4584)
  • [22] R. S. Palais and S. Smale, A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1964), 165-171. MR 0158411 (28:1634)
  • [23] W. Singhof, Minimal coverings of manifolds with balls, Manuscripta Math. 29 (1979), 385-415. MR 545050 (80k:55012)
  • [24] S. Smale, Morse theory and a nonlinear generalization of the Dirichlet problem, Ann. of Math. (2) 80 (1964), 382-396. MR 0165539 (29:2820)
  • [25] F. Takens, The Lusternik-Schnirelmann categories of a product space, Compositio Math. 22 (1970), 175-180. MR 0290365 (44:7549)
  • [26] S. Waner, Equivariant homotopy theory and Milnor's theorem, Trans. Amer. Math. Soc. 258 (1980), 351-368. MR 558178 (82m:55016a)
  • [27] G. W. Whitehead, The homology suspension, Colloq. Topologie Algébrique (Louvain, 1956), pp. 89-95. MR 0094794 (20:1306)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0860382-0
Keywords: Lusternik-Schnirelmann category, abstract critical point theory, homotopy covering, sectional category, strong category
Article copyright: © Copyright 1986 American Mathematical Society

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