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Doctoral Thesis
DOI
10.11606/T.45.2015.tde-02102015-102952
Document
Author
Full name
Vinicius Casteluber Laass
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2015
Supervisor
Committee
Goncalves, Daciberg Lima (President)
Borsari, Lucilia Daruiz
Manzoli Neto, Oziride
Silva, Weslem Liberato
Vendruscolo, Daniel
Title in Portuguese
A propriedade de Borsuk-Ulam para funções entre superfícies
Keywords in Portuguese
Borsuk-Ulam
Grupos de tranças
Superfícies
Abstract in Portuguese
Sejam $M$ e $N$ superfícies fechadas e $\tau: M \to M$ uma involução livre de pontos fixos. Dizemos que uma classe de homotopia $\beta \in [M,N]$ tem a propriedade de Borsuk-Ulam se para toda função contínua $g: M \to N$ que representa $\beta$, existe $x \in M$ tal que $g(\tau(x)) = g(x)$. No caso em que $N$ é diferente de $S^2$ e $RP^2$, mostramos que $\beta$ não ter a propriedade de Borsuk-Ulam é equivalente a existência de um diagrama algébrico envolvendo $\pi_1(M)$, $\pi_1(M_\tau)$, $P_2(N)$ e $B_2(N)$, sendo $M_\tau$ o espaço de órbitas de $\tau$ e sendo $P_2 (N)$ e $B_2(N)$, respectivamente, o grupo de tranças puras e totais de $N$. Para cada caso listado abaixo, nós classificamos todas as classes de homotopia $\beta \in [M,N]$ que têm a propriedade de Borsuk-Ulam: $M = T^2$, $M_\tau = T^2$ e $N = T^2$; $M = T^2$, $M_\tau = K^2$ e $N = T^2$; $M = K^2$ e $N = T^2$; $M = T^2$, $M_\tau = T^2$ e $N = K^2$. No caso em que $N = S^2$, para cada superfície $M$ e involução $\tau: M \to M$, nós classificamos os elementos $\beta \in [M,S^2]$ que têm a propriedade de Borsuk-Ulam. Para fazer tal classificação, nós usamos a teoria de funções equivariantes e a teoria de grau de aplicações. Para classes de homotopia $\beta \in [M,RP^2]$, classificamos aquelas que se levantam para $S^2$. No final, nós consideramos a propriedade de Borsuk-Ulam para ações livres de $Z_p$, com $p$ um inteiro primo positivo. Neste caso, mostramos que se $M$ e $N$ são superfícies fechadas e $Z_p$ age livremente em M, com $p$ ímpar, então sempre existe uma função $f: M \to N$ homotópica a uma função constante e cuja restrição a cada órbita da ação é injetora.
Title in English
The Borsuk-Ulam property for functions between surfaces
Keywords in English
Borsuk-Ulam
Braid groups
Surfaces
Abstract in English
Let $M$ and $N$ be compact surfaces without boundary, and let $\tau: M \to M$ be a fixed-point free involution. We say that a homotopy class $\beta \in [M,N]$ has the Borsuk-Ulam property if for every continuous fuction $g: M \to N$ that represents $\beta$, there exists $x \in M$ such that $g(\tau(x)) = g(x)$. In the case where $N$ is different of $S^2$ and $RP^2$, we show that the fact that $\beta$ does not have the Borsuk-Ulam property is equivalent to the existence of an algebraic diagram involving $\pi_1(M)$, $\pi_1(M_\tau), $P_2(N)$ and $B_2(N)$, where $M_\tau$ is the orbit space of $\tau$ and $P_2(N)$ and $B_2(N) $ are the pure and the full braid groups of the surface $N$ respectively. We then go on to consider the cases of the torus $T^2$ and the Klein bottle $K^2$. Let $M$ and $N$ satisfy one of the following: $M = T^2$, $M_\tau = T^2$ and $N = T^2$; $M = T^2$, $M_\tau = K^2$ and $N = T^2$; $M = K^2$ and $N = T^2$; $M = T^2$, $M_\tau = T^2$ and $N = K^2$. In these cases we classify the homotopy classes $\beta \in [M,N]$ that possess the Borsuk-Ulam property. If $N= S^2$, for every surface $M$ and an involution $\tau: M \to M$, we classify the elements $\beta \in [M, S^2] $ that possess the Borsuk-Ulam property. To obtain this classification, we make use of the theory of equivariant functions and degree theory of maps. For homotopy classes $\beta \in [M,RP^2]$, we classify the classes that admit a lifting to $S^2$. Finally, we consider the Borsuk-Ulam property for free actions of $Z_p$, where $p$ is a prime number. If $M$ and $N$ are compact surfaces without boundary such that $Z_p$ acts freely on $M$, with $p$ odd, we show that there is always a function $f: M \to N$ homotopic to the constant function whose restriction to every orbit of $\tau$ is injective.
 
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Publishing Date
2015-10-08
 
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