In algebraic topology, one studies the group structure of sets of homotopy classes of maps (such as the homotopy groups pin( X)) to obtain information about the spaces in question. It is also possible to place natural topologies on these groups that remember local properties ignored by the algebraic structure. Upon choosing a topology, one is left to wonder how well the added topological structure interacts with the group structure and which results in homotopy theory admit topological analogues. A natural place to begin is to view the n-th homotopy group pi n(X) as the quotient space of the iterated loop space On(X) with the compact-open topology. This dissertation contains a systematic study of these quotient topologies, giving special attention to the fundamental group. The quotient topology is shown to be a complicated and somewhat naive approach to topologizing sets of homotopy classes of maps. The resulting groups with topology capture a great deal of information about the space in question but unfortunately fail to be a topological group quite often. Examples of this failure occurs in the context of a computation, namely, the topological fundamental group of a generalized wedge of circles. This computation introduces a surprising connection to the well-studied free Markov topological groups and indicates that similar failures are likely to appear in higher dimensions. The complications arising with the quotient topology motivate the introduction of well-behaved, alternative topologies on the homotopy groups. Some alternatives are presented, in particular, free topological groups are used to construct the finest group topology on pin(X) such that the map On(X) &rarr; pi n(X) identifying homotopy classes is continuous. This new topology agrees with the quotient topology precisely when the quotient topology does result in a topological group and admits a much nicer theory.
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