Submitted by Guilherme Vituri Fernandes Pinto null (214001018@rc.unesp.br) on 2016-04-11T07:27:24Z No. of bitstreams: 1 Dissertação Guilherme Vituri.pdf: 726432 bytes, checksum: c4db8ed97d1452e129b0f46186ed5a53 (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-04-13T14:34:46Z (GMT) No. of bitstreams: 1 pinto_gvf_me_sjrp.pdf: 726432 bytes, checksum: c4db8ed97d1452e129b0f46186ed5a53 (MD5) / Made available in DSpace on 2016-04-13T14:34:46Z (GMT). No. of bitstreams: 1 pinto_gvf_me_sjrp.pdf: 726432 bytes, checksum: c4db8ed97d1452e129b0f46186ed5a53 (MD5) Previous issue date: 2016-03-18 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo deste trabalho é estudar grande parte do artigo [6], no qual Gottlieb define o subgrupo G(X, x_0) de pi_1(X, x_0) (em que X é um CW-complexo conexo por caminhos), posteriormente chamado de grupo de Gottlieb; o calculamos para diversos espaços, como as esferas, o toro, os espaços projetivos, a garrafa de Klein, etc.; posteriormente, estudamos o artigo [22] de Varadarajan, que generalizou o grupo de Gottlieb para um subconjunto G(A, X) de [A, X]_∗ . Por fim, calculamos G(S^n, S^n). / The goal of this work is to study partially the article [6], in which Gottlieb has defined a subgroup G(X, x_0) of pi_1(X, x_0) (where X is a path-connected CW-complex based at x_0), called "Gottlieb group" in the literature. This group is computed in this work for some spaces, namely the spheres, the torus, the projective spaces, and the Klein bottle. Further, a paper by Varadarajan [22] who has generalized Gottlieb group to a subset G(A, X) of [A, X]_* is studied. Finally, the groups G(S^n, S^n) is computed.

In part one we show that for a compact, metric, locally path-connected topological space X, the shape group of X - as defined in Foundations of Shape Theory by Mardesic and Segal - is isomorphic to the inverse limit of discrete homotopy groups introduced by Conrad Plaut and Valera Berestovskii. We begin by providing the reader preliminary definitions of the fundamental group of a topological space, inverse systems and inverse limits, the Shape Category, discrete homotopy groups, and culminate by providing an isomorphism of the shape and deck groups for peano continua. In part two we develop work and provide further classification of Sharkovskii topological groups, which we call Sharkovskii Groups. We culminate in proving the fact that a locally compact Sharkovskii group must either be the real numbers if it is not compact, or a torsion-free solenoid if it is compact.