Topological obstructions to totally skew embeddings

TL;DR

This paper investigates topological obstructions to totally skew embeddings of smooth manifolds in Euclidean spaces, providing bounds and conjectures related to the minimal embedding dimension based on topological invariants.

Contribution

It introduces new lower bounds for embedding dimensions using topological methods and proposes a conjecture relating embedding dimension to binary digit count of the manifold's dimension.

Findings

Lower bounds closely match known upper bounds in several cases

Evidence supporting the conjecture on embedding dimension involving binary digit count

Revised results with improved understanding of topological obstructions

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).