Book ChapterDOI
Degree for Gradient Equivariant Maps and Equivariant Conley Index
Kazimierz Gęba
- pp 247-272
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TLDR
In this article, the Brouwer degree of a topological invariant with respect to an open bounded subset of a continuous set of subsets of a set of points is defined.Abstract:
In order to establish some notation and terminology we recall that if f : ℝ n → ℝ n is continuous and Ω is an open bounded subset of ℝ n such that f is different from 0 on the boundary of Ω, then there is defined an integer Deg(f, Ω) — the Brouwer (or topological) degree of f with respect to Ω. Obviously, if in the place of all continuos maps and all open bounded subsets of ℝ n we take a smaller class of maps and/or a smaller class of subsets then we may try to define a topological invariant finer then the topological degree.read more
Citations
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Journal ArticleDOI
Pin(2)-Equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture
TL;DR: The Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure was defined in this article.
Journal ArticleDOI
A short treatise on the equivariant degree theory and its applications
TL;DR: In this paper, the authors give a profound introduction to equivariant degree theory, avoiding as far as possible technical details and highly theoretical background, focusing on algebraic, analytical and topological tools for its effective calculation, the latter being illustrated by six concrete examples.
Degree for equivariant gradient maps
TL;DR: This survey article presents result, proved during the recent decade, concerning the degree for equivariant gradient maps, which is shown to be higher than in the past.
Journal ArticleDOI
Global bifurcations of critical orbits of G-invariant strongly indefinite functionals
TL;DR: In this article, the Rabinowitz alternative for G -invariant strongly indefinite functionals was formulated for weak solutions of non-cooperative systems of elliptic differential equations.
Journal ArticleDOI
Existence and continuation of periodic solutions of autonomous Newtonian systems
TL;DR: In this paper, the existence and the continuation of periodic solutions of Newtonian systems are studied and the infinite-dimensional version of the degree for SO (2 ) -equivariant gradient operators defined by the third author in Nonlinear Anal. Math.
References
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Book
Methods of Bifurcation Theory
Shui-Nee Chow,Jack K. Hale +1 more
TL;DR: In this paper, the static and dynamic aspects of bifurcation theory, which are of particular pertinence to differential equations, have been discussed, and a discussion of the changes that occur in the structure of the set of zeros of a function as parameters in the function are varied.
MonographDOI
Isolated Invariant Sets and the Morse Index
TL;DR: On stable properties of the solution set of an ordinary differential equation, see as mentioned in this paper and the Morse index continuuation bibliography for a complete survey of the literature on flow stability and flow properties.
Book
Topology from the differentiable viewpoint
TL;DR: The fundamental theorem of algebra and its application to smooth manifolds and smooth maps was proved by Sard and Brown as discussed by the authors, and the Brouwer degree modulo 2 of a mapping was shown to be equivalent to the Hopf theorem.
Book
Lectures on Algebraic Topology
TL;DR: In this paper, the authors provide a preliminary introduction to the categories, Abelian groups, and homotopy of complexes in the Euclidean space, and a discussion of the application of these categories to Euclideans.