Coupled embeddability

TL;DR

This paper introduces coupled embeddability for maps on product spaces, extending classical embedding theorems and providing new obstructions and examples related to topological and combinatorial properties.

Contribution

It defines coupled embeddability, extends Whitney embedding theorems, and relates these concepts to the $ ext{Z}/2$-coindex, offering new obstructions and nonembeddability results.

Findings

Extended Whitney embedding theorems to coupled embeddability.

Established strong obstructions based on triangulation combinatorics.

Provided examples and nonexamples of coupled embeddings.

Abstract

We introduce the notion of coupled embeddability, defined for maps on products of topological spaces. We use known results for nonsingular biskew and bilinear maps to generate simple examples and nonexamples of coupled embeddings. We study genericity properties for coupled embeddings of smooth manifolds, extend the Whitney embedding theorems to statements about coupled embeddability, and we discuss a Haefliger-type result for coupled embeddings. We relate the notion of coupled embeddability to the $\mathbb{Z}/2$-coindex of embedding spaces, recently introduced and studied by the authors. With a straightforward generalization of these results, we obtain strong obstructions to the existence of coupled embeddings in terms of the combinatorics of triangulations. In particular, we generalize nonembeddability results for certain simplicial complexes to sharp coupled nonembeddability results for certain pairs of simplicial complexes.