On highly regular embeddings

Abstract

A continuous map $\mathbb {R}^d\to \mathbb {R}^N$ is $k$-regular if it maps any $k$ pairwise distinct points to $k$ linearly independent vectors. Our main result on $k$-regular maps is the following lower bound for the existence of such maps between Euclidean spaces, in which $\alpha (k)$ denotes the number of ones in the dyadic expansion of $k$: For $d\geq 1$ and $k\geq 1$ there is no $k$-regular map $\mathbb {R}^d\to \mathbb {R}^N$ for $N<d(k-\alpha (k))+\alpha (k)$. This reproduces a result of Cohen & Handel from 1978 for $d=2$ and the extension by Chisholm from 1979 to the case when $d$ is a power of $2$; for the other values of $d$ our bounds are in general better than Karasev’s (2010), who had only recently gone beyond Chisholm’s special case. In particular, our lower bound turns out to be tight for $k\le 3$. A framework of Cohen & Handel (1979) relates the existence of a $k$-regular map to the existence of a low-dimensional inverse of a certain vector bundle. Thus the non-existence of regular maps into $\mathbb {R}^N$ for small $N$ follows from the non-vanishing of specific dual Stiefel–Whitney classes. This we prove using the general Borsuk–Ulam–Bourgin–Yang theorem combined with a key observation by Hung (1990) about the cohomology algebras of configuration spaces. Our study produces similar lower bound results also for the existence of $\ell$-skew embeddings $\mathbb {R}^d\to \mathbb {R}^N$, for which we require that the images of the tangent spaces of any $\ell$ distinct points are skew affine subspaces. This extends work by Ghomi & Tabachnikov (2008) for the case $\ell =2$.