O Teorema de Borsuk-Ulam: uma versão fraca associada a grupos topológicos / The borsuk-ulam theorem: a weak version associated with topological groups

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Previous issue date: 2017-09-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O Teorema de Borsuk-Ulam clássico afirma que: “Se f : Sn → IRn é uma aplicação contínua, entãoexisteumponto x em Sn talque f(x) = f(−x), ouequivalentemente f(x) = f(A(x)), onde Sn indica a esfera unitária n-dimensional e A : Sn → Sn é a aplicação antipodal”. Se pensamos na superfície terrestre como uma esfera, o caso n = 2 pode ser ilustrado dizendo-se que em cada instante, existe sempre um par de pontos antipodais na superfície da Terra com mesma temperatura e pressão barométrica (supondo que a temperatura e a pressão variam continuamente na superfície). Este trabalho é baseado no artigo “Some generalizations of the Borsuk-Ulam Theorem” de Vendrúsculo, Desideri e Pergher (2011), [8], e tem como principal objetivo apresentar um estudo de uma versão fraca do Teorema de Borsuk-Ulam associada a grupos topológicos. Diz-se que {(X,T);G}, onde X é um espaço topológico equipado por uma involução livre T e G é um grupo topológico, “satisfaz uma versão fraca do Teorema de Borsuk-Ulam”, abreviadamente, “satisfaz WBUT”, se, para cada aplicação contínua f : X → G, temos que o conjunto {x ∈ X; f(x) · f(T(x))−1 ∈ 2G} é diferente do vazio, onde f(T(x))−1 é o simétrico de f(T(x)) em G e 2G = {g ∈ G; g = g−1}. Neste trabalho, relacionamos essa condição fraca com a condição geral de “satisfazer o Teorema de Borsuk-Ulam” (ou “satisfazer BUT”) dada também pelos autores; apresentamos alguns exemplos; considerando G = T2 (toro), detalhamos a demonstração de um resultado que estabelece um critério algébrico para que {(X,T);T2} satisfaça a condição WBUT e de um resultado que dá uma equivalência entre a versão fraca WBUT para triplas {(S,T);T2} e a condição BUT para {(S,T);IR2}, sendo S uma superfície fechada. Por fim, apresentamos um invariante topológico obtido da versão WBUT. Tal invariante, por nós definido, é similar ao obtido da condição BUT e apresentado pelos autores citados. / The classical Borsuk-Ulam Theorem states that: “If f : Sn → IRn is any continuous map, then there exists a point x in Sn such that f(x) = f(−x), or equivalently f(x) = f(A(x)), where Sn denotes the n-dimensional unit sphere and A : Sn → Sn is the antipodal map”. If we think of the Earth’s surface as a sphere, the case n = 2 can be illustrated by saying that at every instant there is always a pair of antipodal points on the Earth’s surface with the same temperature and barometric pressure (assuming that the temperature and pressure vary continuously in the surface). This work is based on the article “Some generalizations of Borsuk-Ulam Theorem” by Ven drúsculo, Desideri and Pergher (2011), [8], and has the main purpose of presenting a study of a weak version of the Borsuk-Ulam Theorem associated with topolog ical groups. It is said that {(X,T);G}, where X is a topological space equipped with a free involution T and G is a topological group “satisfies a Weak version of the Borsuk-Ulam Theorem”, abbreviatedly, “satisfies WBUT” if, given any continuous map f : X → Y , the set {x ∈ X; f(x) · f(T(x))−1 ∈ 2G} is non empty, where f(T(x))−1 is the symmetric of f(T(x)) in G and 2G = {g ∈ G; g = g−1}. In this work, we relate this weak condition with the more general condition of “satisfying the Borsuk-Ulam Theorem” (or “satisfying BUT”) also given by the authors; we present some examples; considering G = T2 (torus), we detail the proof of a result that establishes an algebraic criterion for {(X,T);T2} satisfy the condition WBUT, and of a result that gives an equivalence between the weak version WBUT for triples {(S,T);T2} and the condition BUT for {(S,T);IR2}, where S is a closed surface and T is a free involution on S. Finally, we present a topological invariant obtained from the WBUT version. Such invariant, defined by us, is similar to that obtained from the BUT condition and presented by the cited authors.

Links & Downloads

1. http://hdl.handle.net/11449/152203
2. 000894693
3. 33004153071P0

Tags

Teorema de Borsuk-Ulam

Versão Fraca do Teorema de Borsuk Ulam

Involução Livre

Grupos Topológicos

Superfície

Borsuk-Ulam Theorem

Weak Version of Borsuk-Ulam Theorem

Free Involution

Topological Groups

Surfaces

Additional Fields

Identifer	oai:union.ndltd.org:IBICT/oai:repositorio.unesp.br:11449/152203
Date	25 September 2017
Creators	Marini, Mirela Cristina
Contributors	Universidade Estadual Paulista (UNESP), Fanti, Ermínia de Lourdes Campello, Barbaresco, Évelin Meneguesso
Publisher	Universidade Estadual Paulista (UNESP)
Source Sets	IBICT Brazilian ETDs
Language	Portuguese
Detected Language	English
Type	info:eu-repo/semantics/publishedVersion, info:eu-repo/semantics/masterThesis
Source	reponame:Repositório Institucional da UNESP, instname:Universidade Estadual Paulista, instacron:UNESP
Rights	info:eu-repo/semantics/openAccess
Relation	-1, -1, -1