On Complex Highly Regular Embeddings and the Extended Vassiliev Conjecture

Abstract

A continuous map Cd→ CN is a complex k-regular embedding if any k pairwise distinct points in Cd are mapped by f into k complex linearly independent vectors in CN. The existence of such maps is closely connected with classical problems of algebraic/differential topology, such as embedding/immersion problems. Our central result on complex k-regular embeddings extends results of Cohen & Handel (1978), Chisholm (1979) and Blagojević, Lück & Ziegler (2013) on real k-regular embeddings. We give the following lower bounds for the existence of complex k-regular embeddings. Let p be an odd prime, k≥1 and d= pt for t≥ 1. If there exists a complex k-regular embedding Cd→ CN, then d(k- αp(k)) + αp(k) ≤ N. Here αp(k) denotes the sum of coefficients in the p-adic expansion of k. These lower bounds are obtained by modifying the framework of Cohen & Handel (1978) and a study of Chern classes of complex regular representations. As a main technical result we establish for this an extended Vassiliev conjecture, the following upper bound for the height of the cohomology of an unordered configuration space: If d≥2 and k≥ 2 are integers, and p is an odd prime. Then {equation presented} Furthermore, we give similar lower bounds for the existence of complex l-skew embeddings Cd→ CN, for which we require that the images of the tangent spaces at any l distinct points are skew complex affine subspaces of CN. In addition we give improved lower bounds for the Lusternik-Schnirelmann category of F (Cd, k)/k as well as for the sectional category of the covering F (Cd, k) → F (Cd, k)/k.