Coupled embeddability

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