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Doctoral Thesis
Full name
Vinicius Casteluber Laass
Knowledge Area
Date of Defense
São Paulo, 2015
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
Grupos de tranças
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
Braid groups
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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