If G is a nite group and if X is a G-space, then a Bredon RO(G)-graded equivariantcohomology theory is dened on X. Furthermore, if X is a G-manifold, thereexists a natural Čech hypercohomology theory on X. While Bredon RO(G)-gradedcohomology is important in the theoretical aspects, the Čech cohomology is indispensablewhen computing the cohomology groups. The purpose of this dissertation is toconstruct an isomorphism between these two types of cohomology theories so that theinterplay becomes deeper between the theory and concretely computing cohomologygroups of classical objects. Also, with the aid of Čech cohomology, we can naturallyextend the Bredon cohomology to the more generalized Deligne cohomology.In order to construct such isomorphism, on one hand, we give a new constructionof Bredon RO(G)-graded equivariant cohomology theory from the sheaf-theoreticviewpoint. On the other hand, with Illman's theorem of smooth G-triangulation ofa G-manifold, we extend the existence of good covers from the nonequivariant tothe equivariant case. It follows that, associated to an equivariant good cover of aG-manifold X, there is a bounded spectral sequence converging to Čech hypercohomologywhose E1 page is isomorphic to the E1 page of a Segal spectral sequence whichconverges to the Bredon RO(G)-graded equivariant cohomology. Furthermore, Thisisomorphism is compatible with the structure maps in the two spectral sequences. So there is an induced isomorphism between two limiting objects, which are exactly theČech hypercohomology and the Bredon RO(G)-graded equivariant cohomology.We also apply the above results to real varieties and obtain a quasi-isomorphismbetween two commonly used complexes of presheaves.

If G is a nite group and if X is a G-space, then a Bredon RO(G)-graded equivariantcohomology theory is dened on X. Furthermore, if X is a G-manifold, thereexists a natural Čech hypercohomology theory on X. While Bredon RO(G)-gradedcohomology is important in the theoretical aspects, the Čech cohomology is indispensablewhen computing the cohomology groups. The purpose of this dissertation is toconstruct an isomorphism between these two types of cohomology theories so that theinterplay becomes deeper between the theory and concretely computing cohomologygroups of classical objects. Also, with the aid of Čech cohomology, we can naturallyextend the Bredon cohomology to the more generalized Deligne cohomology.In order to construct such isomorphism, on one hand, we give a new constructionof Bredon RO(G)-graded equivariant cohomology theory from the sheaf-theoreticviewpoint. On the other hand, with Illman's theorem of smooth G-triangulation ofa G-manifold, we extend the existence of good covers from the nonequivariant tothe equivariant case. It follows that, associated to an equivariant good cover of aG-manifold X, there is a bounded spectral sequence converging to Čech hypercohomologywhose E1 page is isomorphic to the E1 page of a Segal spectral sequence whichconverges to the Bredon RO(G)-graded equivariant cohomology. Furthermore, Thisisomorphism is compatible with the structure maps in the two spectral sequences. So there is an induced isomorphism between two limiting objects, which are exactly theČech hypercohomology and the Bredon RO(G)-graded equivariant cohomology.We also apply the above results to real varieties and obtain a quasi-isomorphismbetween two commonly used complexes of presheaves.

Cette thèse concerne principalement deux objets classiques étroitement liés: d'une part la variété des polynômes complexes unitaires de degré $d>1$ à une variable, et à racines simples (donc de discriminant différent de zéro), et d'autre part, les groupes de tresses d'Artin avec d brins. Le travail présenté dans cette thèse propose une nouvelle approche permettant des calculs cohomologiques explicites à coefficients dans n'importe quel faisceau. En vue de calculs cohomologiques explicites, il est souhaitable d'avoir à sa disposition un bon recouvrement au sens de Čech. L'un des principaux objectifs de cette thèse est de construire un tel recouvrement basé sur des graphes (appelés signatures) qui rappellent les `dessins d'enfant' et qui sont associées aux polynômes complexes classifiés par l'espace de polynômes. Cette décomposition de l'espace de polynômes fournit une stratification semi-algébrique. Le nombre de composantes connexes de chaque strate est calculé dans le dernier chapitre ce cette thèse. Néanmoins, cette partition ne fournit pas immédiatement un recouvrement adapté au calcul de la cohomologie de Čech (avec n'importe quels coefficients) pour deux raisons liées et évidentes: d'une part les sous-ensembles du recouvrement ne sont pas ouverts, et de plus ils sont disjoints puisqu'ils correspondent à différentes signatures. Ainsi, l'objectif principal du chapitre 6 est de ``corriger'' le recouvrement de départ afin de le transformer en un bon recouvrement ouvert, adapté au calcul de la cohomologie Čech. Cette construction permet ensuite un calcul explicite des groupes de cohomologie de Čech à valeurs dans un faisceau localement constant. / This thesis mainly concerns two closely related classical objects: on the one hand, the variety of unitary complex polynomials of degree $ d> 1 $ with a variable, and with simple roots (hence with a non-zero discriminant), and on the other hand, the $d$ strand Artin braid groups. The work presented in this thesis proposes a new approach allowing explicit cohomological calculations with coefficients in any sheaf. In order to obtain explicit cohomological calculations, it is necessary to have a good cover in the sense of Čech. One of the main objectives of this thesis is to construct such a good covering, based on graphs that are reminiscent of the ''dessins d'enfants'' and which are associated to the complex polynomials. This decomposition of the space of polynomials provides a semi-algebraic stratification. The number of connected components in each stratum is counted in the last chapter of this thesis. Nevertheless, this partition does not immediately provide a ''good'' cover adapted to the computation of the cohomology of Čech (with any coefficients) for two related and obvious reasons: on the one hand the subsets of the cover are not open, and moreover they are disjoint since they correspond to different signatures. Therefore, the main purpose of Chapter 6 is to ''correct'' the cover in order to transform it into a good open cover, suitable for the calculation of the Čech cohomology. It is explicitly verified that there is an open cover such that all the multiple intersections are contractible. This allows an explicit calculation of cohomology groups of Čech with values in a locally constant sheaf.

Antes da publicação do trabalho An ideal-valued cohomological index theory with applications to Borsuk-Ulam and Bourgin-Yang theorems\"de Fadell e Husseini [20], haviam sido apenas considerados índices numéricos de G-espaços, nos casos G =\'Z IND. 2\' e G um grupo finito. No entanto, tais índices numéricos são obviamente insuficientes no caso de grupos mais complexos, como por exemplo a 1-esfera \'S POT. 1\'. Neste contexto, Fadell e Husseini introduziram o chamado Indice cohomológico de valor ideal: a cada G-espaço X paracompacto, eles associaram um ideal \'Ind POT. G\' (X;K) do anel de cohomología H*(BG;K), onde a cohomologia de Cech H* é considerada com coeficientes em um corpo K e BG é o espaço classificante do grupo G. Além disso, Fadell e Husseini associaram a este ideal o Índice cohomológico de valor numérico, o qual é definido como sendo a dimensão do K-espaço vetorial obtido do quociente entre o anel H*(BG;K) e o ideal \'Ind POT. G\' (X;K). O objetivo principal deste trabalho é apresentar um estudo detalhado deste índice e utilizá-lo no estudo dos resultados sobre grau de aplicações G-equivariantes provados por Hara em \"The degree of equivariant maps\"[24] / Before the appearance of the paper An ideal-valued cohomological index theory with applications to Borsuk-Ulam and Bourgin-Yang theorems\"of Fadell and Husseini [20], had been considered numerical indices of G-spaces, when G = \'Z IND. 2\' and when G is a finite group. However, such numerical indices are obviously insufficient in the case of groups more complexes, for example, G =\'S POT 1\'. In this context Fadell andHusseini, introduced the called valued-ideal cohomological index: to every paracompact G-space X they associated an ideal \'Ind POT. G\' (X,K) of the cohomology ring H*(BG;K), where the Cech cohomology H* is considered with coefficients in a field K and BG is the classifying space of the group G. Moreover, they associated to this ideal the numerical valued cohomological index, that is, the dimension of K-vector space obtained by the quotient between the ring H*(BG;K) and the ideal \'Ind POT. G\' (X,K). The main objective of this work is to present a detailed study of this index and use such index on the study of results on degree of equivariant maps proved by Hara in his paper The degree of equivariant maps\"[24]

O teorema clássico de Borsuk-Ulam nos dá informações à respeito de aplicações \'S POT. n\' \'SETA\' \'R POT. n\', no qual \'S POT. n\' é um \'Z IND. 2\' -espaço livre. O teorema afirma que existe pelo menos uma órbita que é enviada em um único ponto em \'R POT. n\'. Dold [9] estendeu este problema para o contexto de fibrados, considerando aplicações f : S (E) \'SETA\' \'E POT. \'prime\'\' nos quais preservam fibras; aqui, S (E) denota o espaço total do fibrado em esfera sobre B associado ao fibrado vetorial E \'SETA\' B e \'E POT. \'prime\'\' \'SETA\' B é o outro fibrado vetorial. O objetivo desse trabalho é provar esta versão do teorema de Borsuk-Ulam obtida por Dold, chamada versão parametrizada do teorema de Borsuk-Ulam. Nós também provamos uma versão cohomológica deste problema / The classical Borsuk-Ulam Theorem gives information about maps \'S POT. n\' \'ARROW\' \'R POT. n\' where \'S POT. n\' has a free action of the cyclic group \'Z IND. 2\'. The theorem states that there is at least one orbit which is sent to a single point in \'R POT. n\'. Dold [9] extended this problem to a fibre-wise setting, by considering maps f : S (E) \'ARROW\' \' E POT. prime\' which preserve fibres; here, S (E) denotes the total space of the sphere bundle associated over B to a vector bundle E \'ARROW\' B and \'E POT. prime\' \'ARROW\' B is other vector bundle over B. The purpose of this work is to prove this version of the Borsuk-Ulam theorem obtained by A. Dold, called parametrized version of the Borsuk-Ulam theorem. We also prove a cohomological generalization of this problem

Começamos definindo a cohomologia clássica de De Rham e provamos alguns resultados que nos permitem calcular tal cohomologia de algumas variedades diferenciáveis. Com o intuito de provar o Teorema de De Rham, escolhemos fazer a demonstração utilizando a noção de feixes, que se mostra como uma generalização da ideia de cohomologia. Como a cohomologia de De Rham não é a única que se pode definir numa variedade, a questão da unicidade dá origem a teoria axiomática de feixes, que nos dará uma cohomologia para cada feixe dado. Mostraremos que a partir da teoria axiomática de feixes obtemos cohomologias, além das cohomologias clássicas de De Rham, a cohomologia clássica singular e a cohomologia clássica de Cech e mostraremos que essas cohomologias obtidas a partir da noção axiomática são isomorfas as definições clássicas. Concluiremos que se nos restringirmos a apenas variedades diferenciáveis, essas cohomologias são unicamente isomorfas e este será o teorema de De Rham. / We begin by defining De Rhams classical cohomology and we prove some results that allow us a calculation of the cohomology of some differentiable manifolds. In order to prove De Rhams Theorem, we chose to make a demonstration using a notion of sheaves, which is a generalization of the idea of cohomology. Since De Rhams cohomology is not a only one that can be made into a variety, the question of unicity gives rise to axiomatic theory of sheaves, which give us a cohomology for each sheaf given. We will show that from the axiomatic theory of sheaves we obtain cohomologies, besides the classical cohomologies of De Rham, a singular classical cohomology and a classical cohomology of Cech and we will show that cohomologies are obtained from the axiomatic notion are classic definitions. We will conclude that if we restrict ourselves to only differentiable manifolds, these cohomologies are uniquely isomorphic and this will be De Rhams theorem.

O teorema clássico de Borsuk-Ulam nos dá informações à respeito de aplicações \'S POT. n\' \'SETA\' \'R POT. n\', no qual \'S POT. n\' é um \'Z IND. 2\' -espaço livre. O teorema afirma que existe pelo menos uma órbita que é enviada em um único ponto em \'R POT. n\'. Dold [9] estendeu este problema para o contexto de fibrados, considerando aplicações f : S (E) \'SETA\' \'E POT. \'prime\'\' nos quais preservam fibras; aqui, S (E) denota o espaço total do fibrado em esfera sobre B associado ao fibrado vetorial E \'SETA\' B e \'E POT. \'prime\'\' \'SETA\' B é o outro fibrado vetorial. O objetivo desse trabalho é provar esta versão do teorema de Borsuk-Ulam obtida por Dold, chamada versão parametrizada do teorema de Borsuk-Ulam. Nós também provamos uma versão cohomológica deste problema / The classical Borsuk-Ulam Theorem gives information about maps \'S POT. n\' \'ARROW\' \'R POT. n\' where \'S POT. n\' has a free action of the cyclic group \'Z IND. 2\'. The theorem states that there is at least one orbit which is sent to a single point in \'R POT. n\'. Dold [9] extended this problem to a fibre-wise setting, by considering maps f : S (E) \'ARROW\' \' E POT. prime\' which preserve fibres; here, S (E) denotes the total space of the sphere bundle associated over B to a vector bundle E \'ARROW\' B and \'E POT. prime\' \'ARROW\' B is other vector bundle over B. The purpose of this work is to prove this version of the Borsuk-Ulam theorem obtained by A. Dold, called parametrized version of the Borsuk-Ulam theorem. We also prove a cohomological generalization of this problem

Começamos definindo a cohomologia clássica de De Rham e provamos alguns resultados que nos permitem calcular tal cohomologia de algumas variedades diferenciáveis. Com o intuito de provar o Teorema de De Rham, escolhemos fazer a demonstração utilizando a noção de feixes, que se mostra como uma generalização da ideia de cohomologia. Como a cohomologia de De Rham não é a única que se pode definir numa variedade, a questão da unicidade dá origem a teoria axiomática de feixes, que nos dará uma cohomologia para cada feixe dado. Mostraremos que a partir da teoria axiomática de feixes obtemos cohomologias, além das cohomologias clássicas de De Rham, a cohomologia clássica singular e a cohomologia clássica de Cech e mostraremos que essas cohomologias obtidas a partir da noção axiomática são isomorfas as definições clássicas. Concluiremos que se nos restringirmos a apenas variedades diferenciáveis, essas cohomologias são unicamente isomorfas e este será o teorema de De Rham. / We begin by defining De Rhams classical cohomology and we prove some results that allow us a calculation of the cohomology of some differentiable manifolds. In order to prove De Rhams Theorem, we chose to make a demonstration using a notion of sheaves, which is a generalization of the idea of cohomology. Since De Rhams cohomology is not a only one that can be made into a variety, the question of unicity gives rise to axiomatic theory of sheaves, which give us a cohomology for each sheaf given. We will show that from the axiomatic theory of sheaves we obtain cohomologies, besides the classical cohomologies of De Rham, a singular classical cohomology and a classical cohomology of Cech and we will show that cohomologies are obtained from the axiomatic notion are classic definitions. We will conclude that if we restrict ourselves to only differentiable manifolds, these cohomologies are uniquely isomorphic and this will be De Rhams theorem.

Antes da publicação do trabalho An ideal-valued cohomological index theory with applications to Borsuk-Ulam and Bourgin-Yang theorems\"de Fadell e Husseini [20], haviam sido apenas considerados índices numéricos de G-espaços, nos casos G =\'Z IND. 2\' e G um grupo finito. No entanto, tais índices numéricos são obviamente insuficientes no caso de grupos mais complexos, como por exemplo a 1-esfera \'S POT. 1\'. Neste contexto, Fadell e Husseini introduziram o chamado Indice cohomológico de valor ideal: a cada G-espaço X paracompacto, eles associaram um ideal \'Ind POT. G\' (X;K) do anel de cohomología H*(BG;K), onde a cohomologia de Cech H* é considerada com coeficientes em um corpo K e BG é o espaço classificante do grupo G. Além disso, Fadell e Husseini associaram a este ideal o Índice cohomológico de valor numérico, o qual é definido como sendo a dimensão do K-espaço vetorial obtido do quociente entre o anel H*(BG;K) e o ideal \'Ind POT. G\' (X;K). O objetivo principal deste trabalho é apresentar um estudo detalhado deste índice e utilizá-lo no estudo dos resultados sobre grau de aplicações G-equivariantes provados por Hara em \"The degree of equivariant maps\"[24] / Before the appearance of the paper An ideal-valued cohomological index theory with applications to Borsuk-Ulam and Bourgin-Yang theorems\"of Fadell and Husseini [20], had been considered numerical indices of G-spaces, when G = \'Z IND. 2\' and when G is a finite group. However, such numerical indices are obviously insufficient in the case of groups more complexes, for example, G =\'S POT 1\'. In this context Fadell andHusseini, introduced the called valued-ideal cohomological index: to every paracompact G-space X they associated an ideal \'Ind POT. G\' (X,K) of the cohomology ring H*(BG;K), where the Cech cohomology H* is considered with coefficients in a field K and BG is the classifying space of the group G. Moreover, they associated to this ideal the numerical valued cohomological index, that is, the dimension of K-vector space obtained by the quotient between the ring H*(BG;K) and the ideal \'Ind POT. G\' (X,K). The main objective of this work is to present a detailed study of this index and use such index on the study of results on degree of equivariant maps proved by Hara in his paper The degree of equivariant maps\"[24]

Neste trabalho, veremos que a noção de número de Nielsen pode ser estendida para aplicações entre variedades topológicas não necessariamente orientáveis ou compactas, com ou sem fronteira. / In this work, we are going to see that the concept of Nilsen Root Number can be extended to maps between not necessarily orientable nor compact manifolds, with or without boundary.