x, 72 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / The theory of equivariant homology and cohomology was first created by Bredon in his 1967 paper and has since been developed and generalized by May, Lewis, Costenoble, and a host of others. However, there has been a notable lack of computations done. In this paper, a version of the Serre spectral sequence of a fibration is developed for RO ( G )-graded equivariant cohomology of G -spaces for finite groups G . This spectral sequence is then used to compute cohomology of projective bundles and certain loop spaces. In addition, the cohomology of Rep( G )-complexes, with appropriate coefficients, is shown to always be free. As an application, the cohomology of real projective spaces and some Grassmann manifolds are computed, with an eye towards developing a theory of equivariant characteristic classes. / Adviser: Daniel Dugger

Doctor of Philosophy / Department of Mathematics / John S. Maginnis / In this work, we find the module structure of the cohomology of the group of four by four upper triangular matrices (with ones on the diagonal) with entries from the field on three elements modulo its center. Some of the relations amongst the generators for the cohomology ring are also given. This cohomology is found by considering a certain split extension. We show that the associated Lyndon-Hochschild-Serre spectral sequence collapses at the second page by illustrating a set of generators for the cohomology ring from generating elements of the second page. We also consider two other extensions using more traditional techniques. In the first we introduce some new results giving degree four and five differentials in spectral sequences associated to extensions of a general class of groups and apply these to both the extensions.

Sejam G um grupo topológico e X um espaço topológico. Existe uma questão natural associada ao par (G; X) sobre a existência de ações livres e contínuas de G em X. Se tal ação existe, outra questão natural é o estudo de propriedades do espaço de órbitas X / G e, nesse contexto, temos o problema usualmente difícil de se calcular o anel de cohomologia de X / G. Este trabalho é dedicado a essas questões quando X são variedades de Dold P(m;n) especiais e G = Z2. A variedade fechada e suave P(m;n), de dimensão m+2n, é o espaço de órbitas da involução livre T : Sm × CPn → Sm × CPn (x; [z]) → (-x; [ z̄ ]) e foi introduzida por Albrecht Dold em 1956, sendo bastante estudada na literatura e desempenhando papel fundamental na teoria de cobordismo. A principal ferramenta utilizada nesse estudo foi a sequência espectral de Leray-Serre associada à fibração de Borel X → XG → BG; onde XG = (X × EG) / G é a construção de Borel associada ao G-fibrado universal EG → BG. / Let G be a topological group and X be a topological space. There is a natural question associated with the pair (G; X) about the existence of a continuous free action of G on X. If such an action exists, other natural question is the study of properties of the orbit space X / G and, in this setting, the study of the cohomology ring of X / G. This thesis is devoted to these questions when X are special Dold manifolds P(m;n) and G = Z2. The closed smooth (m+2n)-dimensional manifold, P(m;n), is the orbit space of the free involution T : Sm × CPn → Sm × CPn (x; [z]) → (-x; [ z̄ ]) and was introduced by Albrecht Dold in 1956, being well studied in literature and playing a fundamental role in cobordism theory. The main tool used in this study was the Leray-Serre spectral sequence associated with the Borel fibration X → XG → BG; where XG = (X × EG) / G is the Borel construction associated with the universal G-bundle EG → BG.

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Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-01-27Bitstream added on 2014-06-13T20:47:35Z : No. of bitstreams: 1 gomes_nm_me_sjrp.pdf: 689280 bytes, checksum: e326708cb7096ea2640f86118ab525a2 (MD5) / Neste trabalho apresentamos um estudo da sequência espectral associada à uma filtração (finita) de um complexo de cadeias de módulos sobre um anel arbitrário R. Em especial, destacamos as sequências espectrais de Lyndon-Hochschild-Serre e de Cartan-Leray, e algumas aplicações na teoria de homologia. / In this work we present a study of the spectral sequence associated to the filltration (finite) of a chain complex of modules on an arbitrary ring R. In special, we emphasize the spectral sequences of Lyndon-Hochschild-Serre and Cartan-Leray and some applications in the homology theory.

Sejam G um grupo topológico e X um espaço topológico. Existe uma questão natural associada ao par (G; X) sobre a existência de ações livres e contínuas de G em X. Se tal ação existe, outra questão natural é o estudo de propriedades do espaço de órbitas X / G e, nesse contexto, temos o problema usualmente difícil de se calcular o anel de cohomologia de X / G. Este trabalho é dedicado a essas questões quando X são variedades de Dold P(m;n) especiais e G = Z2. A variedade fechada e suave P(m;n), de dimensão m+2n, é o espaço de órbitas da involução livre T : Sm × CPn → Sm × CPn (x; [z]) → (-x; [ z̄ ]) e foi introduzida por Albrecht Dold em 1956, sendo bastante estudada na literatura e desempenhando papel fundamental na teoria de cobordismo. A principal ferramenta utilizada nesse estudo foi a sequência espectral de Leray-Serre associada à fibração de Borel X → XG → BG; onde XG = (X × EG) / G é a construção de Borel associada ao G-fibrado universal EG → BG. / Let G be a topological group and X be a topological space. There is a natural question associated with the pair (G; X) about the existence of a continuous free action of G on X. If such an action exists, other natural question is the study of properties of the orbit space X / G and, in this setting, the study of the cohomology ring of X / G. This thesis is devoted to these questions when X are special Dold manifolds P(m;n) and G = Z2. The closed smooth (m+2n)-dimensional manifold, P(m;n), is the orbit space of the free involution T : Sm × CPn → Sm × CPn (x; [z]) → (-x; [ z̄ ]) and was introduced by Albrecht Dold in 1956, being well studied in literature and playing a fundamental role in cobordism theory. The main tool used in this study was the Leray-Serre spectral sequence associated with the Borel fibration X → XG → BG; where XG = (X × EG) / G is the Borel construction associated with the universal G-bundle EG → BG.

Orientador: Dessislava H. Kochloukova / Tese (doutorado) - Universidade Estadual de Campinas. Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-07T18:15:42Z (GMT). No. of bitstreams: 1 Rodrigues_ClaudenirFreire_D.pdf: 1150293 bytes, checksum: 63045fd15f6ef421699cbcf26de55d92 (MD5) Previous issue date: 2006 / Resumo: Neste trabalho estudamos grupos abstratos finitamente gerados G que são extensões cindidas de um grupo abeliano A por um grupo Q nilpotente de classe 2. Mostramos que se G tem tipo homológico F P3, então o quociente G/N também tem tipo homológico F P3 onde N é o fecho normal do centro de Q em G. Observamos que não existe classificação quando G pode ter tipo FP3, nem classificação para tipo F P2 ou ser finitamente apresentável. Por causa disso nós trabalhamos com um quociente especifico de G. Ainda fica em aberto se cada quociente de G tem tipo FP3 quando G tem tipo FP3. Observamos que isso vale quando G é grupo metabeliano, nesse caso a teoria de Bieri-Strebel pode ser aplicada / Abstract: We study abstract finitely generated groups G that are split extensions from A abelian group by Q nilpotent group of class two. We show that if G has homological type FP3 then the quotient group GjN has homological type FP3 too, where N is the normal closure of the center of Q in G. Since there is no classification when G is of type FP3, nor when G is of type FP2 or finitely presented we work with one specific quotient. It is an open problem whether every quotient of G has type F P3. This holds if G is a metabelian group and in this case the Bieri-Strebel theory applies / Doutorado / Doutor em Matemática

Orientador: Ermínia de Lourdes Campello Fanti / Banca: Luiz Queiroz Pegher / Banca: João Peres Vieira / Resumo: Neste trabalho apresentamos um estudo da sequência espectral associada à uma filtração (finita) de um complexo de cadeias de módulos sobre um anel arbitrário R. Em especial, destacamos as sequências espectrais de Lyndon-Hochschild-Serre e de Cartan-Leray, e algumas aplicações na teoria de homologia. / Abstract: In this work we present a study of the spectral sequence associated to the filltration (finite) of a chain complex of modules on an arbitrary ring R. In special, we emphasize the spectral sequences of Lyndon-Hochschild-Serre and Cartan-Leray and some applications in the homology theory. / Mestre

The Leray-Serre spectral sequence is a fundamental tool for studying singular homology of a fibration E->B with typical fiber F. It expresses H (E) in terms of H (B) and H (F). One of the classic examples of a fibration is given by the free loop space fibration, where the typical fiber is given by the based loop space . The first part of this thesis constructs the Leray-Serre spectral sequence in Morse homology on Hilbert manifolds under certain natural conditions, valid for instance for the free loop space fibration if the base is a closed manifold. We extend the approach of Hutchings which is restricted to closed manifolds. The spectral sequence might provide answers to questions involving closed geodesics, in particular to spectral invariants for the geodesic energy functional. Furthermore we discuss another example, the free loop space of a compact G-principal bundle, where G is a connected compact Lie group. Here we encounter an additional difficulty, namely the base manifold of the fiber bundle is infinite-dimensional. Furthermore, as H ( P) = HF (T P) and H ( Q) =HF (T Q), where HF denotes Floer homology for periodic orbits, the spectral sequence for P -> Q might provide a stepping stone towards a similar spectral sequence defined in purely Floer-theoretic terms, possibly even for more general symplectic quotients. Hutchings’ approach to the Leray-Serre spectral sequence in Morse homology couples a fiberwise negative gradient flow with a lifted negative gradient flow on the base. We study the Morse homology of a vector field that is not of gradient type. The central issue in the Hilbert manifold setting to be resolved is compactness of the involved moduli spaces. We overcome this difficulty by utilizing the special structure of the vector field. Compactness up to breaking of the corresponding moduli spaces is proved with the help of Gronwall-type estimates. Furthermore we point out and close gaps in the standard literature, see Section 1.4 for an overview. In the second part of this thesis we introduce a Lagrangian Floer homology on cotangent bundles with varying Lagrangian boundary condition. The corresponding complex allows us to obtain the Leray-Serre spectral sequence in Floer homology on the cotangent bundle of a closed manifold Q for Hamiltonians quadratic in the fiber directions. This corresponds to the free loop space fibration of a closed manifold of the first part. We expect applications to spectral invariants for the Hamiltonian action functional. The main idea is to study pairs of Morse trajectories on Q and Floer strips on T Q which are non-trivially coupled by moving Lagrangian boundary conditions. Again, compactness of the moduli spaces involved forms the central issue. A modification of the compactness proof of Abbondandolo-Schwarz along the lines of the Morse theory argument from the first part of the thesis can be utilized.