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

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.

14 pages