Otopy classes of equivariant maps

Bartłomiejczyk, Piotr; Gęba, Kazimierz; Izydorek, Marek

doi:10.1007/s11784-010-0013-0

Cited by 16 publications

(25 citation statements)

References 12 publications

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“…10 years later A. Gołębiewska and S. Rybicki [8] generalized this degree to compact Lie groups. The relation between equivariant and equivariant gradient degree theories were studied in [1,2,7].The main goal of this paper is to present a construction and properties of a new degree-type topological invariant Deg ∇ G , which is defined for equivariant gradient perturbations of a equivariant unbounded self-adjoint Hilbert operator with a purely discrete spectrum (in the general case a compact Lie group). As far as we know, the idea of the construction of such an invariant should be attributed to K. Gęba.It is worth pointing out that equivariant gradient perturbations of an equivariant unbounded self-adjoint operator with a purely discrete spectrum appear naturally in a variety of problems in nonlinear analysis, such as the search for periodic solutions of Hamiltonian systems Date: December 24, 2018.…”

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confidence: 99%

Topological degree for equivariant gradient perturbations of an unbounded self-adjoint operator in Hilbert space

Bartłomiejczyk

Kamedulski

Nowak-Przygodzki³

2020

Topology and its Applications

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A. We present a version of the equivariant gradient degree defined for equivariant gradient perturbations of an equivariant unbounded self-adjoint operator with purely discrete spectrum in Hilbert space. Two possible applications are discussed. ITo obtain new bifurcation results, N. Dancer [5] introduced in 1985 a new topological invariant for S 1 -equivariant gradient maps, which provides more information than the usual equivariant one. In 1994 S. Rybicki [14,16] developed the complete degree theory for S 1 -equivariant gradient maps and 3 years later K. Gęba extended this theory to an arbitrary compact Lie group. In 2001 S. Rybicki [15] defined the degree for S 1 -equivariant strongly indefinite functionals in Hilbert space. 10 years later A. Gołębiewska and S. Rybicki [8] generalized this degree to compact Lie groups. The relation between equivariant and equivariant gradient degree theories were studied in [1,2,7].The main goal of this paper is to present a construction and properties of a new degree-type topological invariant Deg ∇ G , which is defined for equivariant gradient perturbations of a equivariant unbounded self-adjoint Hilbert operator with a purely discrete spectrum (in the general case a compact Lie group). As far as we know, the idea of the construction of such an invariant should be attributed to K. Gęba.It is worth pointing out that equivariant gradient perturbations of an equivariant unbounded self-adjoint operator with a purely discrete spectrum appear naturally in a variety of problems in nonlinear analysis, such as the search for periodic solutions of Hamiltonian systems Date: December 24, 2018.

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confidence: 99%

Topological degree for equivariant gradient perturbations of an unbounded self-adjoint operator in Hilbert space

Bartłomiejczyk

Kamedulski

Nowak-Przygodzki³

2020

Topology and its Applications

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“…In the proof of Theorem 1.1 certain properties of the so-called equivariant local maps will be crucial (see [2]). A local G-equivariant gradient map is a pair (f,…”

Section: From Homotopy To Otopymentioning

confidence: 99%

On relations between gradient and classical equivariant homotopy groups of spheres

Gęba

Izydorek

2012

J. Fixed Point Theory Appl.

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Abstract. We investigate relations between stable equivariant homotopy groups of spheres in classical and gradient categories. To this end, the auxiliary category of orthogonal equivariant maps, a natural enlargement of the category of gradient maps, is used. Our result allows for describing stable equivariant homotopy groups of spheres in the category of orthogonal maps in terms of classical stable equivariant groups of spheres with shifted stems. We conjecture that stable equivariant homotopy groups of spheres for orthogonal maps and for gradient maps are isomorphic.

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“…The notion of local maps is introduced in [6] and, independently, in [9]. The relation between gradient and usual local maps (also in the equivariant case) is studied in [2][3][4]. Finally, in [5] authors introduce the topology on the set of local maps and prove that the inclusion of the space of proper maps into the space of local maps is a weak homotopy equivalence if we restrict ourselves to local maps with domains in R + and ranges in R .…”

Section: Introductionmentioning

confidence: 99%