On highly regular embeddings

Abstract

A continuous map Rd → RN 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 α(k) denotes the number of ones in the dyadic expansion of k: For d ≥ 1 and k ≥ 1 there is no k-regular map Rd → RN for N <d(k − α(k)) + α(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 ≤ 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 RN 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 ℓ -skew embeddings Rd → RN, for which we require that the images of the tangent spaces of any ℓ distinct points are skew affine subspaces. This extends work by Ghomi & Tabachnikov (2008) for the case ℓ = 2.