Generalized covering space theories

In this paper, we unify various approaches to generalized covering space theory by introducing a categorical framework in which coverings are defined purely in terms of unique lifting properties. For each category $\mathcal{C}$ of path-connected spaces having the unit disk as an object, we construct a category of $\mathcal{C}$-coverings over a given space $X$ that embeds in the category of $\pi_1(X,x_0)$-sets via the usual monodromy action on fibers. When $\mathcal{C}$ is extended to its coreflective hull $\mathscr{H}(\mathcal{C})$, the resulting category of based $\mathscr{H}(\mathcal{C})$-coverings is complete, has an initial object, and often characterizes more of the subgroup lattice of $\pi_1(X,x_0)$ than traditional covering spaces. We apply our results to three special coreflective subcategories: (1) The category of $\Delta$-coverings employs the convenient category of $\Delta$-generated spaces and is universal in the sense that it contains every other generalized covering category as a subcategory. (2) In the locally path-connected category, we preserve notion of generalized covering due to Fischer and Zastrow and characterize the topology of such coverings using the standard whisker topology. (3) By employing the coreflective hull $\mathbf{Fan}$ of the category of all contractible spaces, we characterize the notion of continuous lifting of paths and identify the topology of $\mathbf{Fan}$-coverings as the natural quotient topology inherited from the path space.Comment: 27 page

Homotopy mapping spaces

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) → 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