Coupled embeddability. Issue 5 (23rd April 2022)

• Record Type: Journal Article
• Authors: Frick, Florian; Harrison, Michael
• Is Part Of: Bulletin of the London Mathematical Society. Volume 54:Issue 5(2022)
• Page Start: 1609
• Page End: 1629
• Publication Date: 2022-04-23
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1112/blms.12646
• Languages: English
• ISSNs: 0024-6093

Abstract

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 Z / 2 $\mathbb{Z}/2$ ‐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.