W. A. Bogley and A. J. Sieradski, Universal path spaces, 5/15/97; revised 2/9/98
Abstract
This paper examines a theory of universal path spaces that properly includes the covering space theory of connected, path connected, semi-locally simply connected spaces. The latter hypothesis that each point in the base space has a relatively simply connected open neighborhood---necessary and sufficient for the existence of a simply connected covering space---is abandoned, thus admitting as base even those spaces that contain arbitrarily small essential loops at wild points. When the base space is a wild metric 2-complex, the universal path space is simply connected if and only if the fundamental group is an omega-group--a group whose elements acquire non-negative real weights and form countable products of all order-type whenever their weights vanish as their appearance in the order-type deepens. Then the endpoint projection enjoys all of the standard lifting properties of covering projections; in particular, the fundamental group of the base space is isomorphic to the group of equivariant self-homeomorphisms of the universal path space. But the properly discontinuous action of the fundamental group on the discrete fibers over the tame points of the base space converge to continuous actions on Cantor fibers (i.e., totally disconnected and perfect ones) over the wild points of the base space. The standard features of covering space theory are thus engulfed by the richer features of universal path space theory for a variety of base spaces whose wild local topology prevents the application of traditional covering space theory.