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.

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.

The main purpose of this thesis is to calculate the integral cohomology ring of the symmetric square of quaternionic projective space, which has been an open problem since computations with symmetric squares were first proposed in the 1930's. The geometry of this particular case forms an essential part of the thesis, and unexpected results concerning two universal Pin(4) bundles are also included. The cohomological computations involve a commutative ladder of long exact sequences, which arise by decomposing the symmetric square and the corresponding Borel space in compatible ways. The geometry and the cohomology of the configuration space of unordered pairs of distinct points in quaternionic projective space, and of the Thom space MPin(4), also feature, and seem to be of independent interest.

Neste trabalho, estudamos a ação dos grupos split metacíclicos \'D IND. (2h+1) POT. 2 nas esferas. Encontramos uma região fundamental dos espaços quocientes, chamados de Formas Espaciais Esféricas Metacíclicas, que foi utilizada para construirmos um conveniente complexo de cadeias destas formas com o qual calculamos o anel de cohomologia e a torção de Reidemeister. Obtivemos também uma relação entre as diferentes torções encontradas / In this work, we study the action of the split metacyclic groups \'D IND. (2h+1) POT. 2 on the spheres. We find a fundamental domain of the quotient spaces, called Metacyclic Spherical Space Forms. Through this region we have built a convenient chain complex of these spaces and we used it to calculate their cohomology ring and Reidemeister torsion. We obtained also a relation between the different torsions found

Neste trabalho, estudamos a ação dos grupos split metacíclicos \'D IND. (2h+1) POT. 2 nas esferas. Encontramos uma região fundamental dos espaços quocientes, chamados de Formas Espaciais Esféricas Metacíclicas, que foi utilizada para construirmos um conveniente complexo de cadeias destas formas com o qual calculamos o anel de cohomologia e a torção de Reidemeister. Obtivemos também uma relação entre as diferentes torções encontradas / In this work, we study the action of the split metacyclic groups \'D IND. (2h+1) POT. 2 on the spheres. We find a fundamental domain of the quotient spaces, called Metacyclic Spherical Space Forms. Through this region we have built a convenient chain complex of these spaces and we used it to calculate their cohomology ring and Reidemeister torsion. We obtained also a relation between the different torsions found

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

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