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 procuramos realizar uma apresentação autocontida sobre os conceitos da teoria algébrica de formas quadráticas e sobre os anéis graduados que surgiram no desenvolvimento desta teoria. Iniciamos procurando esclarecer o sentido da equivalência entre as várias acepções do conceito de forma quadrática. Após a apresentação de ingredientes e resultados geométricos, fazemos um extrato da teoria dos anéis de Witt, conceito que originou a moderna teoria algébrica de formas quadráticas. Disponibilizamos os elementos fundamentais para a formulação das teorias de cohomologia, nos concentrado no desenvolvimento da teoria de cohomologia profinita e, sobretudo, galoisiana. Descrevemos os funtores K0, K1 e K2 da K-teoria clássica e também a K-teoria de Milnor, que é mais adequada para formular questões sobre formas quadráticas. Finalizamos o trabalho com a apresentação de alguns conceitos da Teoria dos Grupos Especiais, uma codificação em primeira-ordem da teoria algébrica das formas quadráticas e exemplificamos sua importância, fornecendo um extrato da prova realizada por Dickmann-Miraglia da conjectura de Marshall sobre assinaturas, que se baseia fortemente nesta teoria. / In this work I try to provide a self-contained presentation on the concepts of algebraic theory of quadratic forms and on the graded rings that have emerged in the development of this theory. I started trying to clarify the meaning of \"equivalence\"between the various meanings of the concept of quadratic form. After the presentation of geometrical ingredients and results, we make an extract of the theory of Witt rings, a concept that originated the modern algebraic theory of quadratic forms. It is provided the key elements for the formulation of cohomology theories, focusing on the development of profinite cohomology theory and, especially, on galoisian cohomology. Are described the functors K0, K1 and K2 of classical K-theory and also the Milnor K-theory, which is more appropriate to formulate questions about quadratic forms. The dissertation is finished with the presentation of some concepts of the Theory of Special Groups, a first-order encoding of algebraic theory of quadratic forms, and with an example its importance by providing an extract of proof by Dickmann-Miraglia of the Marshalls conjecture on signatures, which relies heavily on this theory.

nvestigando a estrutura dos sistemas elementares com simetria de Poincaré em 1 + 1 dimensões, devemos considerar o problema da eliminação das anomalias clássicas, que têm origem no segundo grupo de cohomologia não-trivial deste grupo dinâmico, gerando um termo de Wess-Zumino na ação da partícula relativística. Efetuamos a classificação geral de todos os sistemas elementares em 1 + 1 dimensões, em termos de co-órbitas, mostrando que existe um simplectomorfismo entre o espaço de fase reduzido da partícula e uma determinada co-órbita na álgebra de Lie dual à de Poincaré estendida. / While researching the structure of elementar systems with Poincaré symmetry in 1+1 dimensions, we must be concerned about the problem of elimination of the classical anomalies, which arise from the non-trivial second cohomology group of this dynamical group, generating a Wess-Zumino term in the relativistic particle action. We classify all elementary systems in 1+1 dimensions in terms of co-orbits, showing that there is a symplectomorphism between the reduced phase space of the particle and a certain co-orbit in the Lie algebra dual to the extended Poincaré one.

Sei V ein holomorphes Vektorbündel über einer q-konvexen Mannigfaltigkeit X. Die Andreotti-Grauert-Theorie sagt, dass die r-te Kohomologiegruppe holomorpher Schnitte mit Werten in V endlich-dimensional ist und dass die Kohomologie verschwindet, falls X q-vollständig ist. Ist E ein holomorphes Banachbündel über X, dann ist bekannt, dass die erste Kohomologiegruppe verschwindet, falls X Steinsch ist. Kapitel I gibt einen ausführlichen Überblick über die Arbeit. In Kapitel II wird gezeigt, dass es holomorphe Hilbertbündel über 1-konvexen Mannigfaltigkeiten gibt, für die die erste Kohomologie nicht Hausdorffsch ist. In Kapitel III wird folgender Endlichkeitssatz gezeigt: Ist E ein holomorph triviales Banachbündel oder ein holomorphes Banachbündel von kompaktem Typ mit kompakter Approximationseigenschaft über einer q-konvexen Mannigfaltigkeit X, und ist V ein holomorphes Vektorbündel über X, für das die q-te Kohomologie verschwindet, dann gilt: Die q-te Kohomologie für das Tensorprodukt von V und E ist endlich-dimensional. Ist X q-vollständig, dann verschwindet die r-te Kohomologie, falls r größer oder gleich q ist. Für r größer q kann dies auch für beliebige holomorphe Banachbündel E gezeigt werden. Im Anhang wird skizziert, wie der Ansatz der L2-Methode im Fall r gleich q für Hilbertbündel zu einem Verschwindungssatz führen könnte. / Let V be a holomorphic vector bundle over a q-convex manifold X. The Andreotti-Grauert theory says that the r-th cohomology group of holomorphic section with values in V is finite dimensional and that the cohomology is vanishing if X is q-complete. If E is a holomorphic Banach bundle over X, it is known that the first cohomology group vanishes if X is Stein. Chapter I gives a detailed overview of the work. In chapter II it is shown that there are holomorphic Hilbert bundles over 1-convex manifolds such that the first cohomology of the bundle is not Hausdorff. In chapter III the following finiteness theorem is shown: If E is a holomorphically trivial Banach bundle or a holomorphic Banach bundle of compact type with the compact approximation property over a q-convex manifold X, and if V is a holomorphic vector bundle over X such that the q-th cohomology vanishes, then the following holds true: The q-th cohomology for the tensor product of V and E is finite dimensional. If X is q-complete, then the r-th cohomology vanishes if r is greater or equal q. If r is greater than q, this is shown also for arbitrary holomorphic Banach bundles E. In the appendix it is sketched how for r equal q the L2 method could yield a vanishing theorem for Hilbert bundles.

Neste trabalho estenderemos os resultados obtidos por Hara [34] e J. Jaworowski [38] substituindo as G-variedades por G-variedades generalizadas sobre Z. Além disso, provamos uma fórmula de comparação geral para grau de aplicações de uma variedade generalizada sobre uma esfera que são equivariantes com respeito a ações de grupos finitos, obtendo uma generalização do resultado de A. Kushkuley e Z. Balanov [40] / In this work, we extend the results obtained by Y. Hara [34] and J. Jaworowski [38] by replacing the free G-manifolds by free generalized G-manifolds over Z. Moreover, we prove a general comparison formula for degrees of equivariant maps from a generalized manifold to a sphere which are equivariant with respect to finite group actions, obtaining a generalization of the result of A. Kushkuley and Z. Balanov [40]

Für zufällige dynamische Systeme mit stetiger Zeit existieren zwei wichtige Klassen von Generatoren: Zum einen stationäre zufällige ifferentialgleichungen, i.e. gewöhnliche Differentialgleichungen, die von einem stationärer zufälligen Vektorfeld getrieben werden, und zum anderen stochastische Stratonovichdifferentialgleichungen mit weißem Rauschen. Während die erste Klasse sich gut in den ergodentheoretischen Rahmen der Theorie der zufälligen dynamischen Systeme einfügt, widersetzte sich die zweite Klasse lange Zeit der dynamischen Untersuchung aufgrund des "Konflikts zwischen Ergodentheorie und stochastischer Analysis". In dieser Arbeit wird gezeigt, daß beide Klassen von zufälligen dynamischen Systemen nicht wesentlich verschieden sind, genauer: Zu jeder stochastischen Stratonovichdifferentialgleichung mit weißem Rauschen (unter den üblichen Regularitätsforderungen an die Vektorfelder, die die Existenz von Flüssen garantieren) existiert eine stationäre zufällige Differentialgleichung derart, daß die erzeugten zufälligen dynamischen Systeme konjugiert sind. Als Anwendung wird eine Version des lokalen Linearisierungssatzes von Hartman/Grobman für stochastische Stratonovichdifferentialgleichungen bewiesen. / For continuous time random dynamical systems there exist two important classes of generators: on the one hand stationary random differential quations, i.e. ordinary differential equations driven by a stationary random vector field, and on the other hand stochastic Stratonovich differential equations with white noise. While the first class fits well into the framework of the theory of random dynamical systems, the second class resisted for a long time the dynamical investigation due to the "conflict between ergodic theory and stochastic analysis". The main result of this thesis is that both classes of random dynamical systems are not essentially distinct, more precisely: For each stochastic Stratonovich differential equation with white noise (under usual regularity assumptions) there exists a stationary random differential equation such that the corresponding random dynamical systems are conjugate. As an application a version of the local Hartman/Grobman theorem for stochastic differential equations is proved.

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

Initiée indépendamment par Beilinson et Bernstein et par Brylinski et Kashiwara, l'étude des opérateurs différentiels sur les variétés de drapeaux complets a permis de répondre à une conjecture de Kazhdan et Lusztig. Ayant été poursuivie notamment par les travaux de Borho et Brylinski, cette étude a mis à jour plusieurs propriétés intéressantes sur les opérateurs différentiels sur les variétés de drapeaux. Cependant, en dehors du cas des variétés de drapeaux et du cas des variétés toriques projectives, qui a été étudié de manière combinatoire, les opérateurs différentiels sont plutôt mal compris sur les variétés projectives.Dans cette thèse, nous nous pencherons sur le cas de certaines compactifications magnifiques Y d'espaces symétriques G/H de petit rang, et nous comparerons les résultats obtenus avec ceux connus sur les variétés de drapeaux. Nous allons commencer par construire un opérateur différentiel global sur Y qui ne provient pas de l'action infinitésimale de l'algèbre de Lie de G, ce qui constitue une différence avec le cas des variétés de drapeaux.Ensuite, nous nous intéresserons à trois cas particulier que nous exprimerons comme des quotients GIT d'une certaine grassmannienne X. Grâce à cette description, nous verrons plusieurs similitudes avec le cas des variétés de drapeaux : nous montrerons que l'algèbre des opérateurs globaux sur Y est de type fini, et que pour tout faisceau inversible L sur Y, ses sections globales forment un module simple pour l'algèbre des opérateurs différentiels globaux de Y tordus par L. Enfin, en utilisant des arguments de cohomologie locale, nous montrerons que c'est également le cas pour les groupes de cohomologie supérieurs / Started independently by Beilinson and Bernstein, and by Brylinski and Kashiwara, the study of global differential operators on complete flag varieties has been very useful to answer a conjecture of Kazhdan and Lusztig. In their subsequent work, Borho and Brylinski have discovered many interesting properties on differential operators on flag varieties. But apart from the case of flag varieties, and the case of projective toric varieties, which has been investigated with combinatorial methods, differential operators on projective varieties are rather badly known.In this thesis, we will investigate the case of some wonderful compactifications Y of symmetric spaces G/H of small rank, and we will compare our results with what is known in the case of flag varieties. We will first construct a differential operator on Y which does not come from the infinitesimal action of G, which is different from the case of flag varieties.We will then look at three particular cases, which will be expressed as GIT quotients of some Grassmannian X. With this description, we will find some similarities with the case of flag varieties : we will show that the algebra of global differential operators is of finite type, and that for each invertible sheaf L on Y, the module of its global sections is simple as a module over the algebra of global differential operators of Y twisted by L. Finally, using arguments of local cohomology, we will show that it is still the case for higher cohomology groups

Dada uma ação isométrica livre do grupo binário tetraedral G sobre esferas de dimensão ímpar, obtemos uma decomposição celular finita explícita para as formas espaciais esféricas tetraedrais, fazendo uso do conceito de região (ou domínio) fundamental. A estrutura celular deixa explícita uma descrição do complexo de cadeias sobre o grupo G. Como aplicações, utilizamos o complexo de cadeias e a interpretação geométrica do produto cup para calcular o anel de cohomologia da forma espacial esférica tetraedral em dimensão três, e também calculamos a torção de Reidemeister destes espaços para uma determinada representação de G / Given a free isometric action of a binary tetrahedral group G on odd dimensional spheres, we obtain an explicit finite cellular decomposition of the tetrahedral spherical space forms, using the concept of fundamental domain. The cellular structure gives an explicit description of the associated cellular chain complex over the group G. As applications we use the chain complex and the geometric interpretation of the cup product to calculate the cohomology ring of the tetrahedral spherical space form in three dimension, and also compute the Reidemeister torsion of these spaces for a determined representation of G

Submitted by Bruno Caldeira Carlotti de Souza null (brunoccarlotti@gmail.com) on 2017-03-02T01:45:21Z No. of bitstreams: 1 Dissertação - Bruno C. C. de Souza.pdf: 1030573 bytes, checksum: e3dd1e43953565236359b6d10831067c (MD5) / Approved for entry into archive by LUIZA DE MENEZES ROMANETTO (luizamenezes@reitoria.unesp.br) on 2017-03-07T18:32:31Z (GMT) No. of bitstreams: 1 souza_bcc_me_sjrp.pdf: 1030573 bytes, checksum: e3dd1e43953565236359b6d10831067c (MD5) / Made available in DSpace on 2017-03-07T18:32:31Z (GMT). No. of bitstreams: 1 souza_bcc_me_sjrp.pdf: 1030573 bytes, checksum: e3dd1e43953565236359b6d10831067c (MD5) Previous issue date: 2017-02-23 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo principal deste trabalho é apresentar detalhadamente um estudo sobre um critério, que aparece na referência Coincidence for maps of spaces with finite group action de D. L. Gonçalves, J. Jaworowski, P. L. Q. Pergher e A. Volovikov, para a existência de (H,G)-coincidências de aplicações cujo contradomínio é um CW-complexo finito Y de dimensão k e cujo domínio é um espaço X paracompacto, Hausdorff, conexo e localmente conexo por caminhos e munido de uma G-ação livre, de modo que exista um inteiro m tal que os grupos i-dimensionais de homologia de X sejam triviais nas dimensões 0<i<m e a cohomologia (m+1)-dimensional de G não seja trivial. Para a realização deste estudo foram necessários alguns resultados da teoria de cohomologia de grupos finitos, com ênfase em grupos de cohomologia periódica segundo a teoria de cohomologia de Tate, alguns resultados da teoria de fibrados e algumas noções da teoria de sequências espectrais cohomológicas. / The mais objective of this work is to present in detail a study about a criterion, which appears in the reference Coincidence for maps of spaces with finite group actions by D. L. Gonçalves, J. Jaworowski, P. L. Q. Pergher and A. Volovikov, for existence of (H,G)-coincidences of maps into a finite CW-complex Y with dimension k and whose domain is a paracompact, Hausdorff, connected and locally pathconnected space X with a free action of G, in a way that there exists an integer m such that the ith-homology group of X is trivial for 0<i<m and the (m+1)th-cohomology group of G is nontrivial. For the study of this criterion were needed some results of the theory of cohomology of finite groups, with emphasis on groups with periodic cohomology according to Tate cohomology theory, some results of the theory of fiber bundles and some notions of the theory of cohomological spectral sequences.