Topological obstructions to totally skew embeddings

Abstract

Following Ghomi and Tabachnikov's 2008 work, we study the invariant N(M n) defined as the smallest dimension N such that there exists a totally skew embedding of a smooth manifold M n in ℝ N. This problem is naturally related to the question of estimating the geometric dimension of the stable normal bundle of the configuration space F 2(M n) of ordered pairs of distinct points in M n. We demonstrate that in a number of interesting cases the lower bounds on N(M n) obtained by this method are quite accurate and very close to the best known general upper bound N(M n) ≤ 4n+1 established by Ghomi and Tabachnikov. We also provide some evidence for the conjecture that for every n-dimensional, compact smooth manifold M n (n > 1).