It has been an open question for a long time whether every countable group can be realized as a fundamental group of a compact metric space. Such realizations are not hard to obtain for compact or metric spaces but the combination of both properties turn out to be quite restrictive for the fundamental group. The problem has been studied by many topologists (including Cannon and Conner) but the solution has not been found. In this paper we prove that any countable group can be realized as the fundamental group of a compact subspace of ${\mathbb R}^4$. According to the theorem of Shelah [S. Shelah, Can the Fundamental (Homotopy) Group of a Space be the Rationals?, Proc. Amer. Math. Soc. 103, no. 2, (1988), 627-632] such space cannot be locally path connected if the group is not finitely generated. The theorem is proved by an explicit construction of an appropriate space $X_G$ for every countable group $G$.

COBISS.SI-ID: 16654681

We show that if $G$ is an upper semicontinuous decomposition ${\mathbb R}^n$, $n \ge 4$, into convex sets, then the quotient space ${\mathbb R}^n /G$ is a codimension 1 manifold factor. In particular, we show that ${\mathbb R}^n /G$ has the disjoint arc-disk property.

COBISS.SI-ID: 16618585

We consider open infinite gropes and prove that every continuous map from the minimal grope to another grope is nulhomotopic unless the other grope has a "branch" which is a copy of the minimal grope. Since every grope is the classifying space of its fundamental group, the problem is translated to group theory and a suitable block cancellation of words is used to obtain the result.

COBISS.SI-ID: 16655449