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 $\mathbb {R}^N$. 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 $N(M^n)\leq 4n+1$ established by Ghomi and Tabachnikov. We also provide some evidence for the conjecture that for every $n$-dimensional, compact smooth manifold $M^n$ $(n>1)$, \[ N(M^n)\leq 4n-2\alpha (n)+1.\]