Fundamental groups of locally complicated spaces
by
Gregory R. Conner
Brigham Young University
Coauthors: James W. Cannon, Jack W. Lamoreaux

The notion of fundamental group is very useful in the study of spaces which are locally ``simple'', such as manifolds, simplicial complexes, CW-complexes, and, more generally, spaces which admit a universal cover.

This talk will focus on fundamental groups of spaces which are locally ``complicated''. The standard example of a locally complicated space is the Hawaiian earring (the union of planar circles of radius 1/n, for all naturals n, tangent to the x-axis at the origin.)

The results we will discuss come from four recent preprints and include the following:

1. The realization of the fundamental group of the Hawaiian earring as a group of ``infinite words'', generalizing the notion of free groups of finite rank.
2. The fundamental group of a connected, locally path connected, separable, one-dimensional metric space is countable if and only if it is free, if and only if the space has a universal cover.
3. The fundamental group of a connected, locally path connected, planar set is countable if and only if it is free, if and only if the space has a universal cover.

Date received: April 20, 1998