Root extraction in one-relator groups and slenderness

In this note we strengthen a result of B. B. Newman and use it to prove a conjecture of J. Nakamura stated in \cite{Na} that torsion-free one-relator groups are noncommutatively slender

The fundamental group of reduced suspensions

We classify pointed spaces according to the first fundamental group of their reduced suspension. A pointed space is either of so-called totally path disconnected type or of horseshoe type. These two camps are defined topologically but a characterization is given in terms of fundamental groups. Among totally path disconnected spaces the fundamental group is shown to be a complete invariant for a notion of topological equivalence weaker than that of homeomorphism