A G-Lusternik-Schnirelman category of space with an action of a compact lie group

Marzantowicz, Wacław

doi:10.1016/0040-9383(89)90002-5

Cited by 51 publications

(60 citation statements)

References 7 publications

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“…This concept was introduced in the course of research on the calculus of variations in 1930 [28,23,24]. Extensions of the LS category have been given for actions of compact groups [14,15,29], and for fibrewise spaces [25].…”

Section: Introductionmentioning

confidence: 99%

LS-category of compact Hausdorff foliations

Colman

Hurder

2003

Trans. Amer. Math. Soc.

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Abstract. The transverse (saturated) Lusternik-Schnirelmann category of foliations, introduced by the first author, is an invariant of foliated homotopy type with values in {1, 2, . . . , ∞}. A foliation with all leaves compact and Hausdorff leaf space M/F is called compact Hausdorff. The transverse saturated category cat ∩ | M of a compact Hausdorff foliation is always finite.In this paper we study the transverse category of compact Hausdorff foliations. Our main result provides upper and lower bounds on the transverse category cat ∩ | (M ) in terms of the geometry of F and the Epstein filtration of the exceptional set E. The exceptional set is the closed saturated foliated space which is the union of the leaves with non-trivial holonomy. We prove thatWe give examples to show that both the upper and lower bounds are realized, so the estimate is sharp. We also construct a family of examples for which the transverse category for a compact Hausdorff foliation can be arbitrarily large, though the category of the leaf spaces is constant.

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Section: Introductionmentioning

confidence: 99%

LS-category of compact Hausdorff foliations

Colman

Hurder

2003

Trans. Amer. Math. Soc.

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“…the kernel K of ϕ : G → Homeo(X ) is trivial. Observe that actually this is not a restriction, as then cat G (X ) = cat G/K (X ) [1,11]. If G is finite and X is a manifold, from the properties of finite group actions on X , it follows directly that:…”

Section: Proposition 41 Cat G (A; X ) ≥ Cat(a/g; X/g)mentioning

confidence: 99%

“…The lower ones are standard, while the upper one follows from the main result of [4]. From now on, we shall assume that the action is not free (for free actions the G-category of X coincides with the category of the orbit space X/G, see [11]). …”

Section: Bounds For the Equivariant Categorymentioning

confidence: 99%

“…The question of a computation of equivariant category for a topological space X is of interest, but difficult in general. However, there is a direct, well-known estimate for it, by the category of the orbit space [11].…”

Section: Basic Filtrationmentioning

confidence: 99%

“…Its equivariant version gives a similar estimate for an equivariant C 1 -function f : M → R (cf. [1,3,11]). Consequently, this notion has also important applications in studies of multiplicities of nonlinear variational problems with symmetry (see [1] for a review of results), which can be seen as another justification of our study.…”

Section: Introductionmentioning

confidence: 99%

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Critical points of invariant functions on closed orientable surfaces

Gromadzki

Jezierski²,

Marzantowicz

2014

Bol. Soc. Mat. Mex.

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The relationship between critical points of equivariant functions and topological invariants of an equivariant action on closed manifold is an interesting problem. In this paper, we study this relationship for orientation-preserving actions of finite groups G on a closed orientable surfaces. We give an elementary, but detailed, description of the behaviour of the gradient field of an equivariant C 1 -function, we present an elementary, differential, proof of the Riemann-Hurwitz formula and we construct invariant C 1 -functions with the minimal number of critical orbits. These lead us to show that, with a few exceptions, the equivariant Lusternik-Schnirelmann category of a closed orientable topological surface equals the number of singular orbits of the action.

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Bifurcations of Solutions of SO(2)-Symmetric Nonlinear Problems with Variational Structure

Rybicki¹

Handbook of Topological Fixed Point Theory

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A G-Lusternik-Schnirelman category of space with an action of a compact lie group

Cited by 51 publications

References 7 publications

LS-category of compact Hausdorff foliations

LS-category of compact Hausdorff foliations

Critical points of invariant functions on closed orientable surfaces

Bifurcations of Solutions of SO(2)-Symmetric Nonlinear Problems with Variational Structure

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