Topology studies the geometric properties of spaces that are preserved by continuous deformations. Manifolds are the main examples of topological spaces, with the local properties of Euclidean space in an arbitrary dimension n. They are the higher dimensional analogs of curves and surfaces. For example a circle is a one-dimensional manifold. Balloons and doughnuts are examples of two-dimensional manifolds. A balloon cannot be deformed continuously into a doughnut, so we see that there are essential topological differences between them. An "invariant" of a topological space is a number or an algebraic structure such that topologically equivalent spaces have the same invariant. For example the essential topological difference between the balloon and the doughnut is calculated by the "Euler characteristic", which is 2 for a balloon and 0 for a doughnut. In this thesis I investigate the relation between three different but related invariants of manifolds with dimension divisible by 4: the signature, the Brown-Kervaire invariant and the Arf invariant. The signature invariant takes values in the set (...;-3;-2;-1; 0; 1; 2; 3; ...) of integers. In this thesis we focus on the signature invariant modulo 8, that is its remainder after division by 8. The Brown-Kervaire invariant takes values in the set (0; 1; 2; 3; 4; 5; 6; 7). The Arf invariant takes values in the set (0; 1). The main result of the thesis uses the Brown-Kervaire invariant to prove that for a manifold with signature divisible by 4, the divisibility by 8 is decided by the Arf invariant. The thesis is entirely concerned with pure mathematics. However it is possible that it may have applications in mathematical physics, where the signature modulo 8 plays a significant role.

Orientador: Ermínia de Lourdes Campello Fanti / Banca: Denise de Mattos / Banca: Maria Gorete Carreira Andrade / Resumo: O obejtivo principal deste trabalho é estudar as obstruções "sing" que desempenham papel importante nas demonstrações de certos resultados sobre decomposição de grupos que satisfazem certas hipóteses de dualidade apresentados em [16] e [17], em particular, sobre decomposição de um grupo G adapatada a uma família S de subgrupos de G com (G,S) um par de dualidade de Poincaré. Alguns invariantes cohomológicos e certos resultados envolvendo tais invariantes, decomposição de grupos e/ou grupos e pares de dualidade são também apresentados. / Abstract: The main objective of this work to study the obstructions "sing" which play an important role in demonstrating certain results on the splittings of groups that satisfy certain hypotheses of duality presented in [16] and [17], in particular, the decomposition of a group G adapted to a family S of subgroups of G with (G,S) a Poincaré duality pair. Some cohomological invariants and certain results involving such invariants, a splittings of groups and/or groups and pairs of duality are also presented. / Mestre

Orientador: Ermínia de Lourdes Campello Fanti / Banca: Fernanda Soares Pinto Cardona / Banca: Maria Gorete Carreira Andrade / Resumo: Neste trabalho são abordados alguns aspectos da teoria de dualidade. Ele pode ser dividido em três partes principais. Na primeira demonstramos o teorema de Dualidade de Poincaré para variedades (sem bordo) orientáveis. Para tanto, fez-se necessário o uso do limite direto e cohomologia com suporte compacto. Na segunda definimos grupos de dualidade, em particular, grupo de dualidade de Poincaré, apresentamos alguns resultados e observações sobre a relação existente entre tais grupos e os grupos fundamentais de variedades asféricas fechadas, que é ainda um problema em aberto. Finalmente, alguns resultados envolvendo invariantes cohomológicos "ends" e grupos de dualidade são apresentados. / Abstract: In this work we consider some aspects of duality theory. It can be divided in three principal parts. In the first we prove the Poincaré Duality theorem for orientable manifolds (without boundary). For that, it is necessary the use of the direct limit and cohomology with compact supports. In the second part we de¯ne duality groups, in particular, Poincaré duality groups, we introduce some results and observations about the relationship between such groups and fundamental groups of aspherical closed manifolds, that still is an open problem. Finally, some results envolving the cohomological invariant "ends" and duality groups are presented. / Mestre

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-02-26Bitstream added on 2014-06-13T20:16:04Z : No. of bitstreams: 1 cavalcanti_mps_me_sjrp.pdf: 612728 bytes, checksum: 47d18c69b5ae7b113879890007734ec5 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O obejtivo principal deste trabalho é estudar as obstruções sing que desempenham papel importante nas demonstrações de certos resultados sobre decomposição de grupos que satisfazem certas hipóteses de dualidade apresentados em [16] e [17], em particular, sobre decomposição de um grupo G adapatada a uma família S de subgrupos de G com (G,S) um par de dualidade de Poincaré. Alguns invariantes cohomológicos e certos resultados envolvendo tais invariantes, decomposição de grupos e/ou grupos e pares de dualidade são também apresentados. / The main objective of this work to study the obstructions sing which play an important role in demonstrating certain results on the splittings of groups that satisfy certain hypotheses of duality presented in [16] and [17], in particular, the decomposition of a group G adapted to a family S of subgroups of G with (G,S) a Poincaré duality pair. Some cohomological invariants and certain results involving such invariants, a splittings of groups and/or groups and pairs of duality are also presented.

Made available in DSpace on 2014-06-11T19:30:22Z (GMT). No. of bitstreams: 0 Previous issue date: 2008-03-31Bitstream added on 2014-06-13T19:19:04Z : No. of bitstreams: 1 cellini_cp_me_sjrp.pdf: 781641 bytes, checksum: 70ed1b385d132f8255370c0014be09b4 (MD5) / Neste trabalho são abordados alguns aspectos da teoria de dualidade. Ele pode ser dividido em três partes principais. Na primeira demonstramos o teorema de Dualidade de Poincaré para variedades (sem bordo) orientáveis. Para tanto, fez-se necessário o uso do limite direto e cohomologia com suporte compacto. Na segunda definimos grupos de dualidade, em particular, grupo de dualidade de Poincaré, apresentamos alguns resultados e observações sobre a relação existente entre tais grupos e os grupos fundamentais de variedades asféricas fechadas, que é ainda um problema em aberto. Finalmente, alguns resultados envolvendo invariantes cohomológicos ends e grupos de dualidade são apresentados. / In this work we consider some aspects of duality theory. It can be divided in three principal parts. In the first we prove the Poincaré Duality theorem for orientable manifolds (without boundary). For that, it is necessary the use of the direct limit and cohomology with compact supports. In the second part we de¯ne duality groups, in particular, Poincaré duality groups, we introduce some results and observations about the relationship between such groups and fundamental groups of aspherical closed manifolds, that still is an open problem. Finally, some results envolving the cohomological invariant ends and duality groups are presented.

We study the Poincaré duality map from equivariant Chow cohomology to equivariant Chow groups in the case of torus actions on complete, possibly singular, varieties with isolated fixed points. Our main results yield criteria for the Poincaré duality map to become an isomorphism in this setting. The methods rely on the localization theorem for equivariant Chow cohomology and the notion of algebraic rational cell. We apply our results to complete spherical varieties and their generalizations. / En este artículo estudiamos el homomorfismo de dualidad de Poincaré, el cual relaciona cohomología de Chow equivariante y grupos de Chow equivariante en aquellos casos donde un toro algebraico actúa sobre una variedad singular compacta y con puntos fijos aislados. Nuestros resultados proporcionan criterios bajo los cuales el homomorfismo de dualidadde  Poincaré es un isomorfismo. Para ello, usamos el teorema de localización en cohomología de Chow equivariante y la noción de célula algebraica racional. Aplicamos nuestros resultados a las variedades esféricas compactas y sus generalizaciones.

The irrational rotation algebra A_θ is known to be Poincaré self-dual in the KK-theoretic sense. The spectral triple representing the required K-homology fundamental class was constructed by Connes out of the Dolbeault operator on the 2-torus, but so far, there has not been an explicit description of the dual element. We geometrically construct, for certain elements g of the modular group, a finitely generated projective module L_g over A_θ ⊗ A_θ out of a pair of transverse Kronecker flows on the 2-torus. For upper triangular g, we find an unbounded cycle representing the dual of said module under Kasparov product with Connes' class, and prove that this cycle is invertible in KK(A_θ,A_θ), allowing us to 'untwist' L_g to an unbounded cycle representing the unit for the self-duality of A_θ. / Graduate

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

On associe à chaque variété algébrique définie sur R un complexe de cochaînes filtré, qui calcule la cohomologie à supports compacts et coefficients dans Z_2 de ses points réels. Ce complexe filtré est additif pour les inclusions fermées et acyclique pour la résolution des singularités, et est unique à quasi-isomorphisme filtré près. Il est représenté par la filtration duale de la filtration géométrique sur les chaînes semi-algébriques à supports fermés définie par McCrory and Parusiński, et induit une suite spectrale qui calcule la filtration par le poids sur la cohomologie à supports compacts. Cette suite spectrale est un invariant naturel qui contient les nombres de Betti virtuels. On montre que le produit cartésien de deux variétés nous permet de comparer le produit de leurs complexe de poids et suite spectrale respectifs avec ceux du produit, et on prouve que les produits cap et cup en cohomologie et homologie sont filtrés par rapport à ces filtrations par le poids réelles. / We associate to each algebraic variety defined over R a filtered cochain complex, which computes the cohomology with compact supports and Z_2-coefficients of the set of its real points. This filtered complex is additive for closed inclusions and acyclic for resolution of singularities, and is unique up to filtered quasi-isomorphism. It is represented by the dual filtration of the geometric filtration on semialgebraic chains with closed supports defined by McCrory and Parusiński, and leads to a spectral sequence which computes the weight filtration on cohomology with compact supports. This spectral sequence is a natural invariant which contains the additive virtual Betti numbers. We then show that the cross product of two varieties allows us to compare the product of their respective weight complexes and spectral sequences with those of their product, and prove that the cup and cap products in cohomology and homology are filtered with respect to the real weight filtrations.