The van Kampen obstruction and its relatives

We review a cochain-free treatment of the classical van Kampen obstruction ϑ to embeddability of an n-polyhedron in ℝ²ⁿ and consider several analogs and generalizations of ϑ, including an extraordinary lift of ϑ, which has been studied by J.-P. Dax in the manifold case. The following results are obtained: (1) The mod 2 reduction of ϑ is incomplete, which answers a question of Sarkaria. (2) An odd-dimensional analog of ϑ 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 ϑ is a universal type 1 invariant of singular knots, i.e., knots in ℝ³ 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 ≥ $ \frac{{3(n + 1)}} {2} $ \frac{{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 ℝ2 ^n−k for k < $ \frac{{n - 3}} {2} $ \frac{{n - 3}} {2} , thus extending the Penrose-Whitehead-Zeeman theorem.