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

Clapp, Mónica; Puppe, Dieter

doi:10.1090/S0002-9947-1986-0860382-0

Invariants of the Lusternik-Schnirelmann type and the topology of critical sets
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by Mónica Clapp and Dieter Puppe PDF

Trans. Amer. Math. Soc. 298 (1986), 603-620 Request permission

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).

References

I. Berstein and P. J. Hilton, Category and generalized Hopf invariants, Illinois J. Math. 4 (1960), 437–451. MR 126276
I. Berstein and P. J. Hilton, On suspensions and comultiplications, Topology 2 (1963), 73–82. MR 150775, DOI 10.1016/0040-9383(63)90024-7

The generalized Lusternik-Schnirelmann category of a product space

Tammo tom Dieck, Partitions of unity in homotopy theory, Compositio Math. 23 (1971), 159–167. MR 293625
Tammo tom Dieck, Klaus Heiner Kamps, and Dieter Puppe, Homotopietheorie, Lecture Notes in Mathematics, Vol. 157, Springer-Verlag, Berlin-New York, 1970. MR 0407833

The equivariant Ljusternik-Schnirelmann method for invariant functionals and relative cohomological index theory

Ralph H. Fox, On the Lusternik-Schnirelmann category, Ann. of Math. (2) 42 (1941), 333–370. MR 4108, DOI 10.2307/1968905
T. Ganea, A generalization of the homology and homotopy suspension, Comment. Math. Helv. 39 (1965), 295–322. MR 179791, DOI 10.1007/BF02566956
T. Ganea, Lusternik-Schnirelmann category and strong category, Illinois J. Math. 11 (1967), 417–427. MR 229240
W. J. Gilbert, Some examples for weak category and conilpotency, Illinois J. Math. 12 (1968), 421–432. MR 231375
Brayton Gray, Homotopy theory, Pure and Applied Mathematics, Vol. 64, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. An introduction to algebraic topology. MR 0402714
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362
I. M. James, On category, in the sense of Lusternik-Schnirelmann, Topology 17 (1978), no. 4, 331–348. MR 516214, DOI 10.1016/0040-9383(78)90002-2

Méthodes topologiques dans les problèmes variationels

Michael Mather, Pull-backs in homotopy theory, Canadian J. Math. 28 (1976), no. 2, 225–263. MR 402694, DOI 10.4153/CJM-1976-029-0
Luis Montejano, A quick proof of Singhof’s $\textrm {cat}(M\times S^{1})=\textrm {cat}(M)+1$ theorem, Manuscripta Math. 42 (1983), no. 1, 49–52. MR 693418, DOI 10.1007/BF01171745

Lusternik-Schnirelmann category

A geometric approach

Mitutaka Murayama, On $G$-ANRs and their $G$-homotopy types, Osaka J. Math. 20 (1983), no. 3, 479–512. MR 718960
Richard S. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963), 299–340. MR 158410, DOI 10.1016/0040-9383(63)90013-2
Richard S. Palais, Homotopy theory of infinite dimensional manifolds, Topology 5 (1966), 1–16. MR 189028, DOI 10.1016/0040-9383(66)90002-4
Richard S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115–132. MR 259955, DOI 10.1016/0040-9383(66)90013-9
R. S. Palais and S. Smale, A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1964), 165–172. MR 158411, DOI 10.1090/S0002-9904-1964-11062-4
Wilhelm Singhof, Minimal coverings of manifolds with balls, Manuscripta Math. 29 (1979), no. 2-4, 385–415. MR 545050, DOI 10.1007/BF01303636
S. Smale, Morse theory and a non-linear generalization of the Dirichlet problem, Ann. of Math. (2) 80 (1964), 382–396. MR 165539, DOI 10.2307/1970398
Floris Takens, The Lusternik-Schnirelman categories of a product space, Compositio Math. 22 (1970), 175–180. MR 290365
Stefan Waner, Equivariant homotopy theory and Milnor’s theorem, Trans. Amer. Math. Soc. 258 (1980), no. 2, 351–368. MR 558178, DOI 10.1090/S0002-9947-1980-0558178-7
George W. Whitehead, The homology suspension, Colloque de topologie algébrique, Louvain, 1956, Georges Thone, Liège; Masson & Cie, Paris, 1957, pp. 89–95. MR 0094794