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 Chisholm from 1979 for the case of d being a power of 2; for the other values of d our bounds are in general better than Karasev’s [13], who had only recently gone beyond Chisholm’s special case. In particular, our lower bound turns out to be tight for k ≤ 3. The framework of Cohen and Handel (1979) relates the existence of a k-regular map to the existence of a specific inverse of an appropriate vector bundle. Thus 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 [12] about the cohomology algebras of unordered configuration spaces. Our study produces similar topological 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 and Tabachnikov [8]forl = 2. The details for this work are provided in our paper On highly regular embeddings, Transactions of American Mathematical Society, Published electronically: May 6, 2015, http://dx.doi.org/10.1090/tran/6559.