An obstruction to embedding a simplicial \(n\)-complex into a 2\(n\)-manifold
Let \(f:|K|\to M\) be a continuous map, where \(K\) is a connected \(n\)-dimensional finite simplicial complex and \(M\) is a smooth orientable \(2n\)-manifold without boundary. This paper provides a necessary and sufficient condition for \(f\) to be homotopic to an embedding. Standard techniques imply that \(f\) can be assumed to be a general map, which means, roughly, that \(f\) is a smooth embedding of each simplex, and that \(f\) is globally an embedding except for possible transverse intersection of the images of distinct \(n\)-simplices. Now consider the complex \(J^*K = \{ \sigma \times\tau\mid \sigma \cap \tau =\emptyset\}/(\sigma\times\tau\sim \tau\times\sigma)\). The function which assigns to each \((\sigma,\tau)\) the intersection number of \(f(\sigma)\) and \(f(\tau)\) is shown to be a cocyle. Its cohomology class \(\gamma(f) \in H^{2n}(J^*(K);\mathbb{Z})\) is called the obstruction to finding a homotopy of \(f\) to an embedding. The main result shows that, if \(n\geq 3\), then \(f\) is homotopic to an embedding if and only if \(\gamma(f)=0\).
scientific article; zbMATH DE number 1787586
scientific article