The focus of homotopy theory is often restricted to spaces with some additional structure like CW complexes or manifolds. Such structure provides many tools for computation of homotopy invariants, especially in the case of complexes. The aim of this dissertation is to present various phenomena in homotopy theory which can occur in the situation where such structure is absent. We focus on homotopical smallness and its applications. The first chapter introduces the construction of covering spaces and discusses possible generalizations. We introduce various properties related to the construction and analyze connections between them. The second chapter is devoted to small loops which represent one of the simplest forms of homotopical smallness. We construct a small loop space, i.e. a space in which every loop is small. Accompanying propositions clarify the phenomenon and the reason for its absence in some categories. The extensional classification of small loops provides us a new space SHA which serves as a model for small loops. The third chapter treats homotopical smallness in pointed and unpointed category. Small K-spaces are introduced imitating construction of the small loop space. Classification of homotopical smallness in terms of extension theory generates new examples of spaces which have interesting properties in terms of homotopy theory. The fourth chapter treats homotopical closeness as a generalization of homotopical smallness. It turns out that the notion of closeness is more complicated than that of smallness and harder to study. We present some basic properties and examples of closeness. In the last chapter we provide several examples of spaces whose construction is closely related to homotopical smallness and closeness. Furthermore, we outline the importance of homotopical smallness and closeness in recent realization results.
D.09 Tutoring for postgraduate students
COBISS.SI-ID: 15564377In this invited one-hour plenary lecture at the international conference dedicated to new results in geometry and topology, the work of this project research team was presented, including the most important obtained results.
B.04 Guest lecture
COBISS.SI-ID: 16311641Project leader presented in one-hour lecture at the central research institute for mathematics in Romania the results of this project, which were received with great attention by the specialists.
B.05 Guest lecturer at an institute/university
COBISS.SI-ID: 15867737In this plenary one-hour lecture our recent results were presented on the topics of Mackey groups and topologies, where we are among the leading groups in the world.
B.04 Guest lecture
COBISS.SI-ID: 15968857Our research team member presented at an international conference our results on homotopically Hausdorff spaces, which were obtained during the course of this project, and analyzed the key open problems from this area.
B.04 Guest lecture
COBISS.SI-ID: 15937625