Commutators and squares in free groups

Abstract

Let ${\mathbb{F}}_2$ be the free group generated by $x$ and $y$. In this article, we prove that the commutator of $x^m$ and $y^n$ is a product of two squares if and only if $mn$ is even. We also show using topological methods that there are infinitely many obstructions for an element in ${\mathbb{F}}_2$ to be a product of two squares.