Regular embeddings of manifolds and topology of configuration spaces

TL;DR

This paper explores conditions for the existence of regular embeddings of manifolds and Euclidean spaces, using cohomology obstructions and Stiefel-Whitney classes to establish new lower bounds on embedding dimensions.

Contribution

It introduces new lower bounds on the dimension for regular embeddings of manifolds and Euclidean spaces, utilizing cohomology and characteristic classes.

Findings

Derived cohomology obstructions for regular embeddings.

Established new lower bounds on embedding dimensions.

Utilized Stiefel-Whitney classes to improve bounds for manifolds.

Abstract

For a topological space $X$ we study continuous maps $f : X\to \mathbb R^m$ such that images of every pairwise distinct $k$ points are affinely (linearly) independent. Such maps are called affinely (linearly) $k$-regular embeddings. We investigate the cohomology obstructions to existence of regular embeddings and give some new lower bounds on the dimension $m$ as function of $X$ and $k$, for the cases $X$ is $\mathbb R^n$ or $X$ is an $n$-dimensional manifold. In the latter case, some nonzero Stiefel--Whitney classes of $X$ help to improve the bound.