On the Fundamental Group of Inverse Limits

Abstract

In this paper we study the fundamental group of inverse limits, obtained by upper semi-continuous set valued functions. We present a number of crucial examples which demonstrate the technical difficulties, related to the control of the fundamental group in the inverse limit. Furthermore, these examples realize some important groups as the fundamental groups of inverse limits: free groups and the Hawaiian Earring group. On the other hand, we introduce the right shift of a loop in the inverse limit and prove that the fundamental group of an inverse limit, which is a one-dimensional Peano continuum, is often trivial or uncountable.