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

Abstract

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 \(\mathbb R^5\).