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 $\mathbb{R}<sup>d$ 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 $\mathbb{R}<sup>d$, the nonexistence of nonsingular bilinear maps, and the study of embeddings into $\mathbb{R}<sup>d$ up to isotopy, such as the chirality of spatial graphs.