Small loop spaces

Abstract

The importance of small loops in the covering space theory was pointed out by Brodskiy, Dydak, Labuz, and Mitra in [2] and [3]. A small loop is a loop which is homotopic to a loop contained in an arbitrarily small neighborhood of its base point and a small loop space is a topological space in which every loop is small. Small loops are the strongest obstruction to semi-locally simply connectedness. We construct a small loop space using the Harmonic Archipelago. Furthermore, we define the small loop group of a space and study its impact on covering spaces, in particular its contribution to the fundamental group of the universal covering space.