New results on embeddings of polyhedra and manifolds in Euclidean spaces

The aim of this survey is to present several classical results on embeddings and isotopies of polyhedra and manifolds in $\mathbb R^m$. We also describe the revival of interest in this beautiful branch of topology and give an account of new results, including an improvement of the Haefliger–Weber theorem on the completeness of the deleted product obstruction to embeddability and isotopy of highly connected manifolds in $\mathbb R^m$ (Skopenkov) as well as the unimprovability of this theorem for polyhedra (Freedman, Krushkal, Teichner, Segal, Skopenkov, and Spiez) and for manifolds without the necessary connectedness assumption (Skopenkov). We show how algebraic obstructions (in terms of cohomology, characteristic classes, and equivariant maps) arise from geometric problems of embeddability in Euclidean spaces. Several classical and modern results on completeness or incompleteness of these obstructions are stated and proved. By these proofs we illustrate classical and modern tools of geometric topology (engulfing, the Whitney trick, van Kampen and Casson finger moves, and their generalizations).