Cook continuum, function spaces and free topological groups

A Cook continuum is a one-dimensional compact connected metric space $M$ which is rigid in a very strong sense: for every subcontinuum $C$ of $M$, every continuous map $f$ : $C \to M$ is either the identity or is constant. In my talk, I will show that a Cook continuum can serve as a counterexample to some natural questions in the theory of function spaces as well as in the theory of free topological groups.