The van Kampen obstruction and its relatives

We review a cochain-free treatment of the classical van Kampen obstruction v to embeddability of an n-polyhedron in R-2n and consider several analogs and generalizations of v, including an extraordinary lift of v, which has been studied by J.-P. Dax in the manifold case. The following results are obtained: (1) The mod 2 reduction of I is incomplete, which answers a question of Sarkaria. (2) An odd-dimensional analog of v is a complete obstruction to linkless embeddability (= "intrinsic unlinking") of a given n-polyhedron in R2n+1. (3) A "blown-up" one-parameter version of I 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. (4) Settling a problem of Yashchenko in the metastable range, we find that every PL manifold N nonembeddable in a given R-m, m >= 3(n+1)/2 contains a subset X such that no map N -> R-m sends X and N \ X to disjoint sets. (5) 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 R2n-k for k < n-3/2, thus extending the Penrose-Whitehead-Zeeman theorem.