Jerzy Dydak*
University of Tennessee,
Knoxville, TN, USA
Rips complexes and covers in the uniform category
Berestovskii and Plaut introduced a theory of covers for uniform spaces
generalizing their work for topological groups. The class of uniform spaces for
which their theory works well are so-called coverable spaces. In their approach
composition of covers may not be a cover and it is difficult to determine if a
particular compact space is coverable. Part of the problem is that they
generalize only regular covers in topology and those may not be preserved by
compositions. In this paper we introduce covers for uniform spaces by expanding
the concept of generalized paths of Krasinkiewicz and Minc. We use paths in
Rips complexes and their homotopy classes possess a natural uniform structure.
Applying Rips complexes leads to a natural class of uniform spaces for which
our theory of covers works as well as the classical one, namely the class of
uniformly joinable spaces. In the case of metric continua (compact and
connected metric spaces) that class is identical with pointed 1-movable
spaces, a well-understood class of spaces introduced by shape theorists.
The class of pointed 1-movable continua contains all planar subcontinua
(examples: Hawaiian Earring and the suspension of the Cantor set) and is
preserved by continuous maps. The most notable continuum not being pointed
1-movable is the dyadic solenoid. As an application of our results we present
an exposition of Prajs' homogeneous curve that is path-connected but
not locally connected.
* This is a joint work with N. Brodskiy, B. Labuz, A. Mitra.
