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The Lusternik-Schnirelmann theorem reconsidered. (English) Zbl 0796.57013
Summary: The celebrated theorem of L. Lusternik and L. Schnirelmann [Méthodes topologiques dans les problèmes variationnels (1934; Zbl 0011.02803)], as reformulated by R. S. Palais [Topology 5, 115-132 (1966; Zbl 0143.352)] or J. T. Schwartz [Commun. Pure Appl. Math. 17, 307-315 (1964; Zbl 0152.408)], concerns the behaviour of smooth (i.e., \(C^ 1\)) real-valued functions on a complete Riemannian manifold. The functions are required to be bounded below (or above) and to satisfy one further condition, the Palais-Smale condition. The theorem gives a lower bound, the Lusternik-Schnirelmann category, for the number of critical points of such a function. Without the Palais-Smale condition the result is no longer true. For example the exponential function of a single real variable has no critical points, although it is bounded below and the real line has category one. The purpose of this paper is to establish, by a variant of the classical argument, versions of the Lusternik-Schnirelmann theorem which are valid without the Palais-Smale condition. They reduce to the classical theorem when the manifold is compact and the condition is necessarily satisfied, but in the noncompact case provide new information when the condition is not satisfied.

57R70 Critical points and critical submanifolds in differential topology
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
Full Text: DOI
[1] Berstein, I.; Ganea, T., The category of a map and of a cohomology class, Fund. math., 50, 265-279, (1961/62) · Zbl 0192.29302
[2] Fox, R.H., On the Lusternik-Schnirelmann category, Ann. of math., 42, 333-370, (1941) · JFM 67.0741.02
[3] Freudenthal, H., Uber die enden topologischer Räume und gruppen, Math. Z., 33, 692-713, (1931) · JFM 57.0731.01
[4] Hopf, H., Enden offener Räume und unendliche diskontinuierliche gruppen, Comment. math. helv., 16, 81-100, (1943/44) · Zbl 0060.40008
[5] James, I.M., On category, in the sense of Lusternik-Schnirelmann, Topology, 17, 331-348, (1978) · Zbl 0408.55008
[6] Lusternik, L.; Schnirelmann, L., Méthodes topologiques dans LES problèmes variationnels, (1934), Hermann Paris · JFM 60.1228.04
[7] Palais, R.S., Lusternik-Schnirelmann theory on Banach manifolds, Topology, 5, 115-132, (1966) · Zbl 0143.35203
[8] Palais, R.S.; Smale, S., A generalized Morse theory, Bull. amer. math. soc., 70, 165-172, (1964) · Zbl 0119.09201
[9] Raymond, F., End-point compactification of manifolds, Pacific J. math., 10, 947-963, (1960) · Zbl 0093.37601
[10] Schwartz, J.T., Generalizing the Lusternik-Schnirelmann theory of critical points, Comm. pure appl. math., 17, 307-315, (1964) · Zbl 0152.40801
[11] Siebenmann, L.C., On detecting Euclidean space homotopically among topological manifolds, Invent. math., 6, 245-261, (1968) · Zbl 0169.55201
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