In his 1947 work "Theory of Braids" Emil Artin asked whether the braid group remained unchanged when one considered classes of braids under linkhomotopy, allowing each strand of a braid to pass through itself but not through other strands. We generalize Artin's question to string links over orientable surface M and show that under link-homotopy surface string links form a group PBn(M), which is isomorphic to a quotient of the surface pure braid group PBn(M). Surface braid groups and their properties are an area of active research by González-Meneses, Paris and Rolfsen, Goçalves and Guaschi, and our work explores the geometric and visual beauty of this subject. We compute a presentation of PBn(M) in terms of the generators and relations and discuss the orderability of the group in the case when the surface in question is a unit disk D.

In his 1947 work "Theory of Braids" Emil Artin asked whether the braid group remained unchanged when one considered classes of braids under linkhomotopy, allowing each strand of a braid to pass through itself but not through other strands. We generalize Artin's question to string links over orientable surface M and show that under link-homotopy surface string links form a group PBn(M), which is isomorphic to a quotient of the surface pure braid group PBn(M). Surface braid groups and their properties are an area of active research by González-Meneses, Paris and Rolfsen, Goçalves and Guaschi, and our work explores the geometric and visual beauty of this subject. We compute a presentation of PBn(M) in terms of the generators and relations and discuss the orderability of the group in the case when the surface in question is a unit disk D.

In his 1947 work "Theory of Braids" Emil Artin asked whether the braid group remained unchanged when one considered classes of braids under linkhomotopy, allowing each strand of a braid to pass through itself but not through other strands. We generalize Artin's question to string links over orientable surface M and show that under link-homotopy surface string links form a group PBn(M), which is isomorphic to a quotient of the surface pure braid group PBn(M). Surface braid groups and their properties are an area of active research by González-Meneses, Paris and Rolfsen, Goçalves and Guaschi, and our work explores the geometric and visual beauty of this subject. We compute a presentation of PBn(M) in terms of the generators and relations and discuss the orderability of the group in the case when the surface in question is a unit disk D. / Science, Faculty of / Mathematics, Department of / Graduate

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. / 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.

Em Artin (1925), Artin introduziu o estudo do grupo de tranças, o qual está intimamente relacionado ao estudo de nós e enlaçamentos. Em seu outro artigo Theory of Braids Artin (1947), ele questionou se as noções de isotopia e homotopia de tranças são as mesmas ou diferentes. Tal questão foi respondida muito mais tarde em Goldsmith (1974), onde a autora apresenta um exemplo de trança que é homotópica à trança trivial mas não é equivalente à trança trivial, caracterizando, além disso, o grupo de classes de homotopia de tranças puras no disco como um certo quociente do grupo de tranças puras original. Uma área de pesquisa mais recente nesta teoria é o estudo da ordenação destes grupos de tranças. Em Habegger e Lin (1990) os autores mostram que o grupo de classes de homotopia de tranças puras no disco é nilpotente e livre de torção. Resulta que ele é bi-ordenado. Em Yurasovskaya (2008) a autora fornece uma ordem explícita e calculável para este grupo. Neste trabalho discutiremos e apresentaremos os principais resultados neste contexto. / In Artin (1925), Artin introduced the study of braid groups, which is closely related to the study of knots and links. In his other paper Theory of Braids Artin (1947), he asked if the notions of isotopy and homotopy of braids are different or the same. Such question was answered much later in Goldsmith (1974), where the author presents an example of braid that is homotopic to the trivial braid, but it is not equivalent to the trivial braid, characterizing, beyond that, the group of homotopy classes of braids as an certain quotient of the original braid group. One more recent research area on this theory is the study of ordenation of braid groups. In Habegger e Lin (1990) the authors show that the homotopy group classes of pure braids is nilpotent and torsion free. It follows that it is bi-orderable. In Yurasovskaya (2008) the author provides one explicit and evaluable order for this group. In this work, we will discuss and present the main results involved on this context.

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. / 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.

In his dissertation, F.A. Garside provided a solution to the word and conjugacy problems in the braid group on n-strands, using a particular element that he called the fundamental word. Others have since defined fundamental words in the generalized setting of Artin groups, and even more recently in Garside groups. We consider the problem of finding the number of representations of a power of the fundamental word in these settings. In the process, we find a Pascal-like identity that is satisfied in a certain class of Garside groups.

The theory of infinite multiplication has been studied in the case of the Hawaiian earring group, and has been seen to simplify the description of that group. In this paper we try to extend the theory of infinite multiplication to other groups and give a few examples of how this can be done. In particular, we discuss the theory as applied to symmetric groups and braid groups. We also give an equivalent definition to K. Eda's infinitary product as the fundamental group of a modified wedge product.

Seja $B_mM$ o grupo de tranças com $m$ cordas sobre uma superfície $M$ e seja $N$ uma subsuperfície de $M$. Estudaremos inicialmente condições necessárias e suficientes para as quais $B_nN$ é um subgrupo de $B_mM$ ($m$ podendo ser diferente de $n$), isto é, se considerarmos a inclusão $i\\colon N \\to M$, queremos estabelecer condições sobre $M$ e $N$ para que a aplicação induzida $i_\\ast \\colon B_nN \\to B_mM$ seja injetora. Em seguida, sob certas hipóteses para $N$ e $M$ calcularemos o comensurador, normalizador e centralizador de $B_nN$ em $B_mM$, sendo esse o objetivo principal desta dissertação. / Let $B_m(M)$ be the braid group with $m$ strings on a surface $M$ and let $N$ be a subsurface of $M$. We will study the necessary and sufficient conditions out of which $B_n(N)$ is a subgroup of $B_m(M)$ ($m$ can be different of $n$), it means, if we consider the inclusion $i \\colon N \\to M$, we would like to establish conditions for $M$ and $N$ for the induced application $i_\\ast \\colon B_nN \\to B_mM$ should be injective. After that, under some certain conditions for $M$ and $N$ we will calculate the commensurator, normalizer and centralizer of $Bn(N)$ in $Bm(M)$, being this one the principal objective of this work.

Seja $B_mM$ o grupo de tranças com $m$ cordas sobre uma superfície $M$ e seja $N$ uma subsuperfície de $M$. Estudaremos inicialmente condições necessárias e suficientes para as quais $B_nN$ é um subgrupo de $B_mM$ ($m$ podendo ser diferente de $n$), isto é, se considerarmos a inclusão $i\\colon N \\to M$, queremos estabelecer condições sobre $M$ e $N$ para que a aplicação induzida $i_\\ast \\colon B_nN \\to B_mM$ seja injetora. Em seguida, sob certas hipóteses para $N$ e $M$ calcularemos o comensurador, normalizador e centralizador de $B_nN$ em $B_mM$, sendo esse o objetivo principal desta dissertação. / Let $B_m(M)$ be the braid group with $m$ strings on a surface $M$ and let $N$ be a subsurface of $M$. We will study the necessary and sufficient conditions out of which $B_n(N)$ is a subgroup of $B_m(M)$ ($m$ can be different of $n$), it means, if we consider the inclusion $i \\colon N \\to M$, we would like to establish conditions for $M$ and $N$ for the induced application $i_\\ast \\colon B_nN \\to B_mM$ should be injective. After that, under some certain conditions for $M$ and $N$ we will calculate the commensurator, normalizer and centralizer of $Bn(N)$ in $Bm(M)$, being this one the principal objective of this work.