Some aspects of generalized covering space theory

Abstract

Covering space theory is a classical tool used to characterize the geometry and topology of real or abstract spaces. It seeks to separate the main geometric features from certain algebraic properties. For each conjugacy class of a subgroup of the fundamental group, it supplies a corresponding covering of the underlying space and encodes the interplay between algebra and geometry via group actions. The full applicability of this theory is limited to spaces that are, in some sense, locally simple. However, many modern areas of mathematics, such as fractal geometry, deal with spaces of high local complexity. This has stimulated much recent research into generalizing covering space theory by weakening the covering requirement while maintaining most of the classical utility. This project focuses on the relationships between generalized covering projections, brations with unique path lifting, separation properties of the bers, and continuity of the monodromy.