A lower estimate is proved for the number of critical orbits and critical values of a G-invariant C¹ function , where G is a finite nontrivial group acting freely and orthogonally on . Neither Morse theory nor the minimax method is applied. The proofs are based on a general version of Borsuk’s Antipodal Theorem for equivariant maps of joins of G-sets.
@article{bwmeta1.element.bwnjournal-article-apmv56z2p195bwm,
author = {Danuta Rozp\l och-Nowakowska},
title = {Equivariant maps of joins of finite G-sets and an application to critical point theory},
journal = {Annales Polonici Mathematici},
volume = {57},
year = {1992},
pages = {195-211},
zbl = {0763.57021},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv56z2p195bwm}
}
Danuta Rozpłoch-Nowakowska. Equivariant maps of joins of finite G-sets and an application to critical point theory. Annales Polonici Mathematici, Tome 57 (1992) pp. 195-211. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv56z2p195bwm/
[000] [1] T. Bartsch, Critical orbits of invariant functionals and symmetry breaking, For- schungsschwerpunkt Geometrie, Universität Heidelberg, Heft Nr 34, 1988.
[001] [2] V. Benci and F. Pacella, Morse theory for symmetric functionals on the sphere and an application to a bifurcation problem, Nonlinear Anal. 9 (1985), 763-771. | Zbl 0564.58011
[002] [3] G. Bredon, Introduction to Compact Transformation Groups, Academic Press, 1972. | Zbl 0246.57017
[003] [4] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, 1982. | Zbl 0487.47039
[004] [5] K. Deimling, Nonlinear Functional Analysis, Springer, 1985.
[005] [6] E. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), 139-174. | Zbl 0403.57001
[006] [7] A. Granas and J. Dugundji, Fixed Point Theory, PWN, Warszawa 1982.
[007] [8] M. Hall, The Theory of Groups, Macmillan, New York 1959.
[008] [9] W. Krawcewicz and W. Marzantowicz, Lusternik-Schnirelmann method for functionals invariant with respect to a finite group action, preprint, 1988. | Zbl 0703.58012
[009] [10] W. Krawcewicz and W. Marzantowicz, Some remarks on the Lusternik-Schnirelmann method for non-differentiable functionals invariant with respect to a finite group action, preprint, 1988. | Zbl 0728.58005
[010] [11] J. Milnor, Construction of universal bundles, II, Ann. of Math. 63 (1956), 430-436. | Zbl 0071.17401
[011] [12] R. S. Palais, Critical point theory and the minimax principle, in: Proc. Sympos. Pure Math. 15, Amer. Math. Soc., 1970, 185-212. | Zbl 0212.28902
[012] [13] R. S. Palais, Lusternik-Schnirelmann theory on Banach manifolds, Topology 5 (1966), 115-132. | Zbl 0143.35203
[013] [14] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conf. Ser. in Math. 65, Amer. Math. Soc., 1984.
[014] [15] V. A. Rokhlin and D. B. Fuks, An Introductory Course in Topology. Geometric Chapters, Nauka, Moscow 1977 (in Russian). | Zbl 0417.55002
[015] [16] J. A. Wolf, Spaces of Constant Curvature, University of California, Berkeley, Calif., 1972.