Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2004. / Includes bibliographical references (p. 87-89).

The symmetric automorphism group of a free product is a group rich in algebraic structure and with strong links to geometric configuration spaces. In this thesis I describe in detail and for the first time the (co)homology of the symmetric automorphism groups. To this end I construct a classifying space for the Fouxe-Rabinovitch automorphism group, a large normal subgroup of the symmetric automorphism group. This classifying space is a moduli space of 'cactus products', each of which has the homotopy type of a wedge product of spaces. To study this space we build a combinatorial theory centred around 'diagonal complexes' which may be of independent interest. The diagonal complex associated to the cactus products consists of the set of forest posets, which in turn characterise the homology of the moduli spaces of cactus products. The machinery of diagonal complexes is then turned towards the symmetric automorphism groups of a graph product of groups. I also show that symmetric automorphisms may be determined by their categorical properties and that they are in particular characteristic of the free product functor. This goes some way to explain their occurence in a range of situations. The final chapter is devoted to a class of configuration spaces of Euclidean n-spheres embedded disjointly in (n+2)-space. When n = 1 this is the configuration space of unknotted, unlinked loops in 3-space, which has been well studied. We continue this work for higher n and find that the fundamental groups remain unchanged. We then consider the homology and the higher homotopy groups of the configuration spaces. Our last contribution is an epilogue which discusses the place of these groups in the wider field of mathematics. It is the functoriality which is important here and using this new-found emphasis we argue that there should exist a generalised version of the material from the final chapter which would apply to a far wider range of configuration spaces.

Quasicrystal is the term used for a solid that possesses an essentially discrete diffraction pattern without having translational symmetry. Compared to periodic crystals, this difference in structure gives quasicrystals new properties that make them interesting to study -- both from a mathematical and from a physical point of view. In this thesis we review a mathematical description of quasicrystals that aims at generalizing the well-established theory of periodic crystals. We see how this theory can be connected to the cohomology of groups and how we can use this connection to classify quasicrystals. We also review an experimental method, NIXSW (Normal Incidence X-ray Standing Waves), that is ordinarily used for surface structure determination of periodic crystals, and show how it can be used in the study of quasicrystal surfaces. Finally, we define the reduced lattice and show a way to plot lattices in MATLAB. We see that there is a connection between the diffraction pattern and the reduced lattice and we suggest a way to describe this connection.

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-03-15Bitstream added on 2014-06-13T19:47:19Z : No. of bitstreams: 1 castro_fr_me_sjrp.pdf: 783980 bytes, checksum: fd80e9aa8c69641da08ee43dfa94509d (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo principal deste trabalho é estudar a Teoria de Cohomologia de Grupos visando apresentar de forma detalhada algumas aplicações dessa teoria na Topologia e na Algebra, mais especificamente na Teoria de Grupos, com destaque para o Teorema de Schur-Zassenhaus e o Teorema de Classificação de p-grupos que possuem um subgrupo ciclico de índice p (p primo). / The aim of this work is to study the Cohomology Theory of Groups in order to present in detailed form some applications of this theory in Topology and in Algebra, more specifically, in the Theory of Groups, with prominence for the Schur-Zassenhaus Theorem and the Theorem of Classification of p-groups which contain a cyclic subgroup of index p, where p is a prime.

Orientador: Ermínia de Lourdes Campello Fanti / Banca: Oziride Manzoli Neto / Banca: Maria Gorete Carreira Andrade / Resumo: Para todo grupo G infinito, finitamente gerado, pode-se obter para o invariante algébrico "end", mais precisamente o número de ends e(G), uma fórmula cohomológica 1-dimensional. O principal objetivo deste trabalho é apresentar, sob certas hipóteses, uma fórmula cohomológica 1-dimensional para o invariante algébrico e(G,H), definido por Scott e Houghton, onde H é um subgrupo de G (Teorema de Swarup). Para tanto, o conceito de subconjunto H-quase invariante de G e resultados como a interpretação do grupo de cohomologia H1(G,M) em termos de derivações (à direita), onde M é um ZG-módulo, e o Lema de Shapiro, são resultados imprescindíveis. Algumas relações desses invariantes com ends de espaços são também apresentadas. / Abstract: For all infinite group G, finitely generated, one can obtain for the algebric invariant "end", more precisely the number of ends e(G), a cohomological 1-dimensional formula. The main objective of this work is to present, under certain hypotheses, a cohomological 1-dimensional formula for the algebric invariant e(G,H), defined by Scott and Houghton, where H is a subgroup of G (Swarup's Theorem). In order to do so, the concept of subset H-almost invariant of G and results like the interpretation of the cohomological group H1(G,M) in terms of derivations (to the right), where M is a ZG-module, and the Shapiro's Lemma, are fundamental results. Some relations of these invariants with space ends are also presented. / Mestre

Orientador: Ermínia de Lourdes Campello Fanti / Banca: Luiz Queiroz Pergher / Banca: Maria Gorete Carreira Andrade / Resumo: O objetivo principal deste trabalho é estudar a Teoria de Cohomologia de Grupos visando apresentar de forma detalhada algumas aplicações dessa teoria na Topologia e na Algebra, mais especificamente na Teoria de Grupos, com destaque para o Teorema de Schur-Zassenhaus e o Teorema de Classificação de p-grupos que possuem um subgrupo ciclico de índice p (p primo). / Abstract: The aim of this work is to study the Cohomology Theory of Groups in order to present in detailed form some applications of this theory in Topology and in Algebra, more specifically, in the Theory of Groups, with prominence for the Schur-Zassenhaus Theorem and the Theorem of Classification of p-groups which contain a cyclic subgroup of index p, where p is a prime. / Mestre

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

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 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-04-12Bitstream added on 2014-06-13T19:55:23Z : No. of bitstreams: 1 santos_ap_me_sjrp.pdf: 749833 bytes, checksum: 8be58c6f81e3ac600ff8f26430348533 (MD5) / Para todo grupo G infinito, finitamente gerado, pode-se obter para o invariante algébrico end, mais precisamente o número de ends e(G), uma fórmula cohomológica 1-dimensional. O principal objetivo deste trabalho é apresentar, sob certas hipóteses, uma fórmula cohomológica 1-dimensional para o invariante algébrico e(G,H), definido por Scott e Houghton, onde H é um subgrupo de G (Teorema de Swarup). Para tanto, o conceito de subconjunto H-quase invariante de G e resultados como a interpretação do grupo de cohomologia H1(G,M) em termos de derivações (à direita), onde M é um ZG-módulo, e o Lema de Shapiro, são resultados imprescindíveis. Algumas relações desses invariantes com ends de espaços são também apresentadas. / For all infinite group G, finitely generated, one can obtain for the algebric invariant end, more precisely the number of ends e(G), a cohomological 1-dimensional formula. The main objective of this work is to present, under certain hypotheses, a cohomological 1-dimensional formula for the algebric invariant e(G,H), defined by Scott and Houghton, where H is a subgroup of G (Swarup's Theorem). In order to do so, the concept of subset H-almost invariant of G and results like the interpretation of the cohomological group H1(G,M) in terms of derivations (to the right), where M is a ZG-module, and the Shapiro's Lemma, are fundamental results. Some relations of these invariants with space ends are also presented.

Nous présentons une étude de la cohomologie à coefficients tordus de la réalisation géométrique des systèmes de liaison. Plus précisément, si (S, Ƒ, ℒ) est un groupe fini p-local, nous travaillons sur la cohomologie H*(\ℒ\, M) de la réalisation géométrique de ℒ, avec un Z(p)[π₁(\ℒ\)]-module M en coefficients, et ses liens avec les éléments Fᶜ-stables H* (Ƒᶜ, M) ⊆ H*(S, M) à travers l’inclusion de BS dans \ℒ\. Après avoir donné la définition des éléments Ƒᶜ-stables, nous étudions l’endomorphisme de H*(S, M) induit par un (S, S)-bi-ensemble Ƒᶜ-caractéristique et nous montrons que sous certaine hypothèse et si l’action est nilpotent, alors on a un isomorphisme naturel H*(\ℒ\, M) ≌ H* (Ƒᶜ,M). Ensuite, nous regardons les actions p-résolubles à travers la notion de sous-groupe p-local d’index premier à p ou une puissance de p. Nous montrons que si l’action de π₁(\ℒ\) sur M se factorise par un p'-groupe alors on a aussi un isomorphisme naturel. Pour une action p-résoluble plus général, nous obtenons un résultat dans le cas des systèmes réalisables. Ces résultats nous conduisent à la conjecture qu’on a un isomorphisme naturel pour tout groupe fini p-local et toute action p-résoluble. Nous donnons quelque outils pour étudier cette conjecture. Nous travaillons sur les produits de groupes finis p-locaux avec la formule de Kunneth et les systèmes de liaison que se décomposent bien vis-à-vis de la suite exacte longue de Mayer-Vietoris. Finalement, nous étudions les sous-groupes essentiels d’un produit couronné par Cp. Nous finissons par des exemples qui soulignent, qu’en général, on ne peut espérer un isomorphisme entre H*(\ℒ\, M) et H*(Ƒᶜ, M). / The aim of this work is to study the cohomology with twisted coefficients of the geometric realization of linking systems. More precisely, if (S, Ƒ, ℒ) is a p-local finite group, we work on the cohomology H*(\ℒ\, M) of the geometric realization of ℒ with coefficients in a Z(p)[π₁(\ℒ\)]-module M and its links with the Ƒᶜ-stables H*(Ƒᶜ, M) ⊆ H*(S, M) trough the inclusion of BS in \ℒ\. After we give the definition of Ƒᶜ-stable elements , we study the endomorphism of H*(S, M) induced by an Fc-characteristic (S, S)-biset and we show that, if the action is nilpotent- and we assume an hypothesis, we have a natural isomorphism H*(\ℒ\, M) ≌ H* (Fᶜ;M). Secondly, we look at p-solvable actions of π₁(\ℒ\) on M through the notion of p-local subgroups of index a power of p or prime to p. If the action factors through a p'-group, we show that there si also a natural isomorphism. We then work on extending this to any-p-solvable action and we get some positive answer then the p-local finite groupis realizable. Theses leads to the conjecture that it is true for any-p-local finite group and any-p-solvable actions. We also give some tools to study this conjecture on examples. We look at products of p-local finite groups with Kunneth Formula and linking system which can be decomposed in a way which behaves well with Mayer-Vietoris long exact sequence. Finally, we study essential subgroups of wreath productsby Cp. We finish with some examples which illustrate that, in general, we cannot hope an isomorphism between H*(\ℒ\, M) and H*(Ƒᶜ, M).