On definable subgroups of the fundamental group

摘要

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 [23] and [7]. A compactness theorem for the fundamental group of a Peano continuum is given. 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. (C) 2019 Elsevier B.V. All rights reserved.