Free maps in critical dimension on low-dimensional tori and closed surfaces

We present a method to build free immersions in critical dimension on $m$-tori for $m=2,3,4,5$ by using a factorization trick inspired by tori immersions in critical dimension. As an application, we show that the set of smooth free maps from a closed surface $M$ to ${\mathbb{R}}^5$ is nonempty. In particular, every closed surface embeds freely in ${\mathbb{R}}^5$.