Obstructions to approximating maps of \(n\)-manifolds into \(\mathbb{R}^{2n}\) by embeddings
The first result, which is proved in one paragraph, is that if \(n\geq 3\), then a general position map \(f\) from a closed \(n\)-manifold \(K\) into \({\mathbb R}^{2n-1}\) is the projection of an embedding into \({\mathbb R}^{2n}\) if and only if the composite \(K\to{\mathbb R}^{2n-1} \hookrightarrow{\mathbb R}^{2n}\) is approximable by embeddings. The main result involves the case \(n=2\). If \(K\) is a closed 2-manifold and \(f:K\to {\mathbb R}^3\) is a general position map, an obstruction \(\beta(f)\) to approximability of the composite \(K\to{\mathbb R}^3\hookrightarrow\mathbb{R}^4\) by embeddings is defined, using resolvability of triple points. Certain quadratic forms \(q_T\) associated to the intersection form on \(H^1(K;{\mathbb Z}_2)\) are defined, and it is proved that \(\beta(f)\) equals the product of the Arf invariants of the forms \(q_T\).
scientific article; zbMATH DE number 1801745
scientific article