Seja p um \'S POT. 3\' - fibrado principal sobre uma esfera \'S POT. n\' , com n \' >OU=\' 4 . O objetivo deste trabalho é calcular os tipos de homotopia do grupo de gauge \'G IND. p\' desses fibrados p, estendendo o resultado determinado por A. Kono [25] quando n = 4. Apresentamos fórmulas explícitas para o operador bordo na seqüência exata de homotopia associada com a aplicação avaliação ev : m(\'S POT. n\' , B \'S POT. 3\' ) \'SETA\' B \'S POT. 3\' , traduzindo o problema nos cálculos envolvendo grupos de homotopia de esferas. Calculamos todos os casos clássicos, ou seja, aqueles que podem ser avaliados usando as informações encontradas no livro de H. Toda [46], determinando o tipo de homotopia do grupo de gauge desses fibrados para cada n \' > OU =\' 25 / Let p be a principal \'S POT. 3\' - bundle over a sphere \'S POT. n\' , with n\' > or =\' 4\'. The subject of this work is to calculate the homotopy type of the gauge group \'G IND. p\' of these bundles p, extending the result determined by A. Kono [25] when n = 4. We present explicit formulas for the boundary operator in the homotopy exact sequence associated with the evaluation map ev : m(\'S POT. n\' , B \'S POT. 3\' ) \' ARROW\' B \'S POT. 3\' , translating that problem into calculations involving homotopy groups of sphere. We calculate all the classical cases, namely those that can be dealt with using the information in the book of H. Toda [46], determining the homotopy type of the gauge group of these bundles for each n \'> or = 25

Seja p um \'S POT. 3\' - fibrado principal sobre uma esfera \'S POT. n\' , com n \' >OU=\' 4 . O objetivo deste trabalho é calcular os tipos de homotopia do grupo de gauge \'G IND. p\' desses fibrados p, estendendo o resultado determinado por A. Kono [25] quando n = 4. Apresentamos fórmulas explícitas para o operador bordo na seqüência exata de homotopia associada com a aplicação avaliação ev : m(\'S POT. n\' , B \'S POT. 3\' ) \'SETA\' B \'S POT. 3\' , traduzindo o problema nos cálculos envolvendo grupos de homotopia de esferas. Calculamos todos os casos clássicos, ou seja, aqueles que podem ser avaliados usando as informações encontradas no livro de H. Toda [46], determinando o tipo de homotopia do grupo de gauge desses fibrados para cada n \' > OU =\' 25 / Let p be a principal \'S POT. 3\' - bundle over a sphere \'S POT. n\' , with n\' > or =\' 4\'. The subject of this work is to calculate the homotopy type of the gauge group \'G IND. p\' of these bundles p, extending the result determined by A. Kono [25] when n = 4. We present explicit formulas for the boundary operator in the homotopy exact sequence associated with the evaluation map ev : m(\'S POT. n\' , B \'S POT. 3\' ) \' ARROW\' B \'S POT. 3\' , translating that problem into calculations involving homotopy groups of sphere. We calculate all the classical cases, namely those that can be dealt with using the information in the book of H. Toda [46], determining the homotopy type of the gauge group of these bundles for each n \'> or = 25

<p>Isotropy representations provide powerful tools for understanding the classification of equivariant principal bundles over the $2$-sphere. We consider a $\Gamma$-equivariant principal $G$-bundle over $S^2$ with structural group $G$ a compact connected Lie group, and $\Gamma \subset SO(3)$ a finite group acting linearly on $S^2.$ Let $X$ be a topological space and $\Gamma$ be a group acting on $X.$ An isotropy subgroup is defined by $\Gamma_x = \{\gamma \in \Gamma \lvert \gamma x=x\}.$ Assume $X$ is a $\Gamma$-space and $A$ is the orbit space of $X$. Let $\varphi: A\rightarrow X$ be a continuous map with $\pi \circ \varphi = 1_A$. An isotropy groupoid is defined by $\mathfrak{I} = \{(\gamma,a) \in \Gamma\times A \lvert \ \gamma \in \Gamma_{\varphi(a)}\}.$ An isotropy representation of $\mathfrak{I}$ is a continuous map $\iota : \mathfrak{I} \rightarrow G$ such that the restriction map $\mathfrak{I}_a \rightarrow G$ is a group homomorphism. $\Gamma$- equivariant principal $G$-bundles are studied in two steps; \begin{enumerate} [1)] \item the restriction of an equivariant bundle to the $\Gamma$ equivariant 1-skeleton $X \subset S^2$ where $\mathfrak{I}$ is isotropy representation of $X$ over singular set of the $\Gamma$-sets in $S^2$ \item the underlying $G$-bundle $\xi$ over $S^2$ determined by $c(\xi)\in \pi_2(BG).$ \end{enumerate}</p> / Master of Science (MSc)

This  thesis  consists  of two papers  and  a summary.  The  papers  both  deal with  affine algebraic complex  varieties,  and  in particular such  varieties  in dimension  three  that have a non-trivial action  of one of the  one-dimensional  algebraic  groups  Ga   :=  (C, +) and  Gm  :=  (C*, ·).  The methods  used  involve  blowing up  of subvarieties, the correspondances between  Ga - and  Gm - actions  on an affine variety  X with locally nilpotent derivations  and Z-gradings  respectively  on O(X) and passing from a filtered algebra  A to its associated graded  algebra  gr(A). In Paper  I, we study  Russell’s hypersurface  X , i.e. the affine variety  in the affine space A4 given by the equation  x + x2y + z3 + t2 = 0. We reprove by geometric means Makar-Limanov’s result which states  that X is not isomorphic to A3 – a result which was crucial to Koras-Russell’s proof of the linearization conjecture  for Gm -actions on A3. Our method consist in realizing X as an open part  of a blowup M  −→ A3 and to show that each Ga -action on X descends to A3 . This follows from considerations of the graded  algebra  associated to O(X ) with respect  to a certain filtration. In Paper  II, we study  Ga-threefolds X  which have  as their  algebraic  quotient  the  affine plane  A2  = Sp(C[x, y]) and  are a principal  bundle  above the  punctured plane  A2  :=  A2 \ {0}. Equivalently, we study  affine Ga -varieties  Pˆ  that extend  a principal  bundle  P over A2, being P together  with an extra  fiber over the origin in A2. First  the trivial  bundle  is studied,  and some examples of extensions  are given (including  smooth  ones which are not isomorphic  to A2 × A). The  most  basic among  the  non-trivial  principal  bundles  over A2 is SL2 (C)  −→ A2, A  1→  Ae1 where e1  denotes  the first unit  vector,  and we show that any non-trivial  bundle  can be realized as a pullback  of this  bundle  with  respect  to  a morphism  A2  −→ A2. Therefore  the  attention is then  restricted to extensions  of SL2(C)  and  find two families of such extensions  via a study of the  graded  algebras  associated  with  the  coordinate  rings  O(Pˆ)  '→ O(P ) with  respect  to  a filtration  which is defined in terms  of the Ga -actions  on P and Pˆ  respectively.

Já é bem estabelecido na geometria diferencial o uso de fibrados principais com grupo de estru- tura para a definição e o estudo de algumas estruturas geométricas na base do fibrado. O uso de fibrados principais com grupoide de estrutura na definição de estruturas geométricas sobre varieda- des não tem sido muito explorada. O único exemplo do uso desses fibrados para definir estruturas geométricas foi dado Haefliger. Ele mostrou que folheações regulares sobre uma variedade estão em correspondência com uma classe de fibrados principais com grupoide de estrutura, e usando a classificação de fibrados principais ele obtive a classificação de folheações regulares a menos de homotopia sobre uma variedade aberta. Neste trabalho propomos uma definição a qual generaliza as folheações regulares para produzir uma classe de fibrados vetoriais ancorados e provamos para eles um teorema de classificação no espirito do teorema de Haefliger. Depois aplicamos a teoria desenvolvida aos grupoides com formas multiplicativas e mostramos como a nossa definição per- mite trasladar a geometria guardada na forma multiplicativa para a base do fibrado principal. Em seguida voltamos para o caso de folheações regulares e mostramos que a nossa proposta permite incluir novas estruturas transversais à folheação. / It is well know in differencial geometry the use of principal bundles with structure group to define and study some geometric structures on the base of the bundle. The use of principal bun- dle with a structure groupoid has not been extensively studied yet. The only example using this kind of bundle was provided by Haefliger in his study of regular foliations. Haefliger showed that regular foliations can be identified with some class of principal bundles with structure groupoid, then by using the classifying theorem of principal bundles he arrived to the classification theorem of regular foliations up to homotopy on open manifolds. In this work we will propose a definition that generalizes regular foliations to include anchored vector bundles and, will prove a classification theorem for these structures in the spirit of Haefligers theorem. Then we will apply this theory to groupoids with multiplicative forms and show that our definition permits to transfer the geometry encoded in the multiplicative form to the base of the bundle. Then we will back to the case of regular foliations and show that our proposal allow new transversal structures to the foliation.

Já é bem estabelecido na geometria diferencial o uso de fibrados principais com grupo de estru- tura para a definição e o estudo de algumas estruturas geométricas na base do fibrado. O uso de fibrados principais com grupoide de estrutura na definição de estruturas geométricas sobre varieda- des não tem sido muito explorada. O único exemplo do uso desses fibrados para definir estruturas geométricas foi dado Haefliger. Ele mostrou que folheações regulares sobre uma variedade estão em correspondência com uma classe de fibrados principais com grupoide de estrutura, e usando a classificação de fibrados principais ele obtive a classificação de folheações regulares a menos de homotopia sobre uma variedade aberta. Neste trabalho propomos uma definição a qual generaliza as folheações regulares para produzir uma classe de fibrados vetoriais ancorados e provamos para eles um teorema de classificação no espirito do teorema de Haefliger. Depois aplicamos a teoria desenvolvida aos grupoides com formas multiplicativas e mostramos como a nossa definição per- mite trasladar a geometria guardada na forma multiplicativa para a base do fibrado principal. Em seguida voltamos para o caso de folheações regulares e mostramos que a nossa proposta permite incluir novas estruturas transversais à folheação. / It is well know in differencial geometry the use of principal bundles with structure group to define and study some geometric structures on the base of the bundle. The use of principal bun- dle with a structure groupoid has not been extensively studied yet. The only example using this kind of bundle was provided by Haefliger in his study of regular foliations. Haefliger showed that regular foliations can be identified with some class of principal bundles with structure groupoid, then by using the classifying theorem of principal bundles he arrived to the classification theorem of regular foliations up to homotopy on open manifolds. In this work we will propose a definition that generalizes regular foliations to include anchored vector bundles and, will prove a classification theorem for these structures in the spirit of Haefligers theorem. Then we will apply this theory to groupoids with multiplicative forms and show that our definition permits to transfer the geometry encoded in the multiplicative form to the base of the bundle. Then we will back to the case of regular foliations and show that our proposal allow new transversal structures to the foliation.

This Master's thesis relates to continuum mechanics and its connection with selected directions of modern differential geometry, which deal with geometric structures and objects. These are mainly tensors, bundles, varieties and jets. The first part is devoted to the mechanics of the continuum itself and its description in several areas, others deal with mathematical concepts and their possible application in mechanics.