On definable subgroups of the fundamental group

TL;DR

This paper advances the understanding of the fundamental group in path connected metric spaces, providing new theorems, characterizations, and results on its structure, compactness, and decomposition.

Contribution

It offers several new theorems, including a compactness result for Peano continua and a characterization of the shape kernel, strengthening prior results in the field.

Findings

Strengthened main theorems from previous work.

Established a compactness theorem for the fundamental group of Peano continua.

Showed that free decompositions in certain spaces are nonconstructive.

Abstract

We present several new theorems concerning the first fundamental group of a path connected metric space. Among the results proven are strengthenings of the main theorems of \cite{Sh2} and \cite{CoCo}. A compactness theorem for the fundamental group of a Peano continuum is given. A useful characterization for the shape kernel of a locally path connected space is presented, along with a very succinct proof of the fact that for such a space the Spanier and shape kernel subgroups coincide (see \cite{BF}). We also show that a free decomposition of the fundamental group of a locally path connected Polish space cannot be nonconstructive. Numerous other results and examples illustrating the sharpness of our theorems are provided.