@MISC{Melikhov09thevan, author = {Sergey A. Melikhov}, title = {The van Kampen obstruction and its relatives}, year = {2009} }

Share

OpenURL

Abstract

We review a cochain-free treatment of the classical van Kampen obstruction ϑ to embeddability of an n-polyhedron into R2n and consider several analogues and generalizations of ϑ, including an extraordinary lift of ϑ which in the manifold case has been studied by J.-P. Dax. The following results are obtained. • The mod2 reduction of ϑ is incomplete, which answers a question of Sarkaria. • An odd-dimensional analogue of ϑ is a complete obstruction to linkless embeddability (=“intrinsic unlinking”) of the given n-polyhedron in R2n+1. • A “blown up ” 1-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. • Settling a problem of Yashchenko in the metastable range, we obtain that every PL manifold N, non-embeddable in a given Rm, m ≥ 3(n+1), contains a subset X 2 such that no map N → Rm 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 R2n−k for k < n−3 extending the Penrose–Whitehead–Zeeman theorem.