New results on embeddings of polyhedra and manifolds in Euclidean spaces

The aim of this survey is to present several classical results on embedding s and isotopies of polyhedra and manifolds in R-m. We also describe the rev ival of interest in this beautiful branch of topology and give an account o f new results, including an improvement of the Haefliger-Weber theorem on t he completeness of the deleted product obstruction to embeddability and iso topy of highly connected manifolds in R-m (Skopenkov) as well as the unimpr ovability of this theorem for polyhedra (freedman, Krushkal, Teichner, Sega l, Skopenkov, and Spiel) and for manifolds without the necessary connectedn ess assumption (Skopenkov). We show how algebraic obstructions (in terms of cohomology, characteristic classes, and equivariant maps) arise from geome tric problems of embeddability in Euclidean spaces. Several classical and m odern results on completeness or incompleteness of these obstructions are s tated 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).