Spaces of embeddings: Nonsingular bilinear maps, chirality, and their generalizations

TL;DR

This paper investigates the topology of embedding spaces of a given space into Euclidean space using combinatorial triangulations, providing bounds related to nonembeddability, nonsingular bilinear maps, and chirality.

Contribution

It introduces a combinatorial formula for upper bounds on sphere dimensions that map into embedding spaces, extending previous results on nonembeddability and chirality.

Findings

Derived bounds for embedding dimensions

Extended nonembeddability results

Connected chirality and bilinear map studies

Abstract

Given a space X we study the topology of the space of embeddings of X into $\mathbb{R}^d$ through the combinatorics of triangulations of X. We give a simple combinatorial formula for upper bounds for the largest dimension of a sphere that antipodally maps into the space of embeddings. This result summarizes and extends results about the nonembeddability of complexes into $\mathbb{R}^d$, the nonexistence of nonsingular bilinear maps, and the study of embeddings into $\mathbb{R}^d$ up to isotopy, such as the chirality of spatial graphs.