The cohomology ring of a finite cyclic group was explicitly computed by Cartan and Eilenberg in their 1956 book on Homological Algebra. It is surprising that the cohomology ring for the next simplest example, that of a finite abelian group, has still not been treated in a systematic way. The results that we do have are combinatorial in nature and have been obtained using "brute force" computations. In this thesis we will give a systematic method for computing the cohomology ring of a finite abelian group. A major ingredient in this treatment will be the Tate resolution of a commutative ring R (with trivial group action) over the group ring RG, for some finite abelian group G. Using the Tate resolution we will be able to compute the cohomology ring for a finite cyclic group, and confirm that this computation agrees with what is known from Cartan-Eilenberg. Then we will generalize this technique to compute the cohomology ring for a finite abelian group. The presentation we will give is simpler than what is in the literature to date. We will then see that a straightforward generalization of the Tate resolution from a group ring to an arbitrary ring defined by monic polynomials will yield a method for computing the Hochschild cohomology algebra of that ring. In particular we will re-prove some results from the literature in a much more unified way than they were originally proved. We will also be able to prove some new results.

In 2003, Johannes Kellendonk and Ian Putnam introduced pattern equivariant cohomology for tilings. In 2006, Betseygail Rand defined a type of pattern equivariant cohomology that incorporates rotational symmetry, using representation of the rotation group. In this doctoral thesis we study the relationship between these two types of pattern equivariant cohomology, showing exactly how to calculate one from the other in the case in which the rotation group is a finitely generated abelian group of free rank 1. We apply our result by calculating the cohomology of the pinwheel tiling.

The cohomology ring of a finite cyclic group was explicitly computed by Cartan and Eilenberg in their 1956 book on Homological Algebra. It is surprising that the cohomology ring for the next simplest example, that of a finite abelian group, has still not been treated in a systematic way. The results that we do have are combinatorial in nature and have been obtained using "brute force" computations. In this thesis we will give a systematic method for computing the cohomology ring of a finite abelian group. A major ingredient in this treatment will be the Tate resolution of a commutative ring R (with trivial group action) over the group ring RG, for some finite abelian group G. Using the Tate resolution we will be able to compute the cohomology ring for a finite cyclic group, and confirm that this computation agrees with what is known from Cartan-Eilenberg. Then we will generalize this technique to compute the cohomology ring for a finite abelian group. The presentation we will give is simpler than what is in the literature to date. We will then see that a straightforward generalization of the Tate resolution from a group ring to an arbitrary ring defined by monic polynomials will yield a method for computing the Hochschild cohomology algebra of that ring. In particular we will re-prove some results from the literature in a much more unified way than they were originally proved. We will also be able to prove some new results.

Neste trabalho, estudamos a cohomologia de de Rham e métodos para os seus cálculos. Finalizamos com aplicações da cohomologia de de Rham em teoremas da topologia / In this work we study the de Rham cohomology and methods for its calculations. We close it with applications of the Rham cohomology in theorems from topology

Neste trabalho consideramos o problema de enumerar G-fibrados sobre variedades de dimensão baixa (menor ou igual a 3), em particular fibrados vetoriais sobre as ?spherical space forms? tridimensionais. É dada uma resposta completa para estas questões e na seção 5.1 são colocadas tabelas que explicitam os possíveis fibrados vetoriais sobre as ?spherical space forms?. Este tipo de problema é recorrente em topologia algébrica e por motivos dados pela teoria de homotopia, é preciso calcular certos invariantes algébricos com sistemas de coeficientes locais, o que torna o problema mais interessante. Mostramos ainda que sobre condições consideravelmente abrangentes no grupo estrutural G, os G-fibrados sobre variedades de dimensão menor ou igual a três podem ser enumerados de maneira efetiva / In this work we consider the problem of enumerating G-bundles over low dimensional manifolds (dimension · 3) and in particular vector bundles over the three dimensional ?spherical space forms?. We give a complete answer to these questions and in section 5.1 we give tables for the possible vector bundles over the ?spherical space forms?. We deal with the problem of enumerating vector bundles over a class of manifolds. This is a long standing classical problem in algebraic topology, and because of homotopy theoretical reasons, it implies calculations of algebraic invariants with local system of coefficients, and thus becomes a cumbersome target away from the trivial occurrences. Although, we show that, under reasonably wide assumptions on the structure group G, G-bundles over low (lower or equal to three) dimensional manifolds can be counted effectively

Neste trabalho consideramos o problema de enumerar G-fibrados sobre variedades de dimensão baixa (menor ou igual a 3), em particular fibrados vetoriais sobre as ?spherical space forms? tridimensionais. É dada uma resposta completa para estas questões e na seção 5.1 são colocadas tabelas que explicitam os possíveis fibrados vetoriais sobre as ?spherical space forms?. Este tipo de problema é recorrente em topologia algébrica e por motivos dados pela teoria de homotopia, é preciso calcular certos invariantes algébricos com sistemas de coeficientes locais, o que torna o problema mais interessante. Mostramos ainda que sobre condições consideravelmente abrangentes no grupo estrutural G, os G-fibrados sobre variedades de dimensão menor ou igual a três podem ser enumerados de maneira efetiva / In this work we consider the problem of enumerating G-bundles over low dimensional manifolds (dimension · 3) and in particular vector bundles over the three dimensional ?spherical space forms?. We give a complete answer to these questions and in section 5.1 we give tables for the possible vector bundles over the ?spherical space forms?. We deal with the problem of enumerating vector bundles over a class of manifolds. This is a long standing classical problem in algebraic topology, and because of homotopy theoretical reasons, it implies calculations of algebraic invariants with local system of coefficients, and thus becomes a cumbersome target away from the trivial occurrences. Although, we show that, under reasonably wide assumptions on the structure group G, G-bundles over low (lower or equal to three) dimensional manifolds can be counted effectively

No description available.

We discuss the Cauchy problem for the Dolbeault cohomology in a domain of C n with data on a part of the boundary. In this setting we introduce the concept of a Carleman function which proves useful in the study of uniqueness. Apart from an abstract framework we show explicit Carleman formulas for the Dolbeault cohomology.

The action of various operations on the cohomology of nilpotent Lie algebras is studied. In the cohomology of any Lie algebra, we show that the existence of certain nontrivial compositions of higher cohomology operations implies the existence of hypercube-like structures in cohomology, which in turn establishes the Toral Rank Conjecture for that Lie algebra. We provide examples in low dimensions and exhibit an infinite family of nilpotent Lie algebras of arbitrary dimension for which such structures exist. A new proof of the Toral Rank Conjecture is also given for free two-step nilpotent Lie algebras.

This dissertation describes an algorithm and its implementation in the computer algebra system GAP for constructing the cohomology ring and Ext-algebra for certain group algebras kG. We compute in the Morita equivalent basic algebra B of kG and obtain the cohomology ring and Ext-algebra for the group algebra kG up to isomorphism. As this work is from a computational point of view, we consider the cohomology ring and Ext-algebra via projective resolutions.There are two main methods for computing projective resolutions. One method uses linear algebra and the other method uses noncommutative Grobner basis theory. Both methods are implemented in GAP and results in terms of timings are given. To use the noncommutative Grobner basis theory, we have implemented and designed an alternative algorithm to the Buchberger algorithm when given a finite dimensional algebra in terms of a basis consisting of monomials in the generators of the algebra and action of generators on the basis.The group algebras we are mainly concerned with here are for simple groups in characteristic dividing the order of the group. We have computed the Ext-algebra and cohomology ring for a variety of simple groups to a given degree and have thus added many more examples to the few that have thus far been computed.