The van Kampen obstruction and its relatives
  We review a cochain-free treatment of the classical van Kampen obstruction \theta to embeddability of an n-polyhedron into R^{2n} and consider several analogues and generalizations of \theta, including an extraordinary lift of \theta which in the manifold case has been studied by J.-P. Dax. The following results are obtained.   - The mod 2 reduction of \theta is incomplete, which answers a question of Sarkaria.   - An odd-dimensional analogue of \theta is a complete obstruction to linkless embeddability (="intrinsic unlinking") of the given n-polyhedron in R^{2n+1}.   - A "blown up" 1-parameter version of \theta is a universal type 1 invariant of singular knots, i.e. knots in R^3 with a finite number of rigid transverse double points. We use it to decide in simple homological terms when a given integer-valued type 1 invariant of singular knots admits an integral arrow diagram (=Polyak-Viro) formula.   - Settling a problem of Yashchenko in the metastable range, we obtain that every PL manifold N, non-embeddable in a given R^m, m\ge 3(n+1)/2, contains a subset X such that no map N\to R^m sends X and N-X to disjoint sets.   - We elaborate on McCrory's analysis of the Zeeman spectral sequence to geometrically characterize "k-co-connected and locally k-co-connected" polyhedra, which we embed in R^{2n-k} for k<(n-3)/2 extending the Penrose-Whitehead-Zeeman theorem. 
Sergey A. Melikhov