Topological obstructions to totally skew embeddings

Abstract: Following Ghomi and Tabachnikov we study topological obstructions to totally skew embeddings of a smooth manifold M in Euclidean spaces. 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) of ordered pairs of distinct points in M. We demonstrate that in a number of interesting cases the lower bounds obtained by this method are quite accurate and very close to the best known general upper bound. We also provide some evidence for the conjecture that each n-dimensional, compact smooth manifold M^n (n>1), admits a totally skew embedding in the Euclidean space of dimension N = 4n-2alpha(n)+1 where alpha(n)=number of non-zero digits in the binary representation of n. This is a revised version of the paper (accepted for publication in A.M.S. Transactions).