The van Kampen Obstruction and Its Relatives

We review a cochain-free treatment of the classical van Kampen obstruction $\vartheta$ to embeddability of an $n$-polyhedron in $\mathbb R^{2n}$ and consider several analogs and generalizations of $\vartheta$, including an extraordinary lift of $\vartheta$, which has been studied by J.-P. Dax in the manifold case. The following results are obtained:
(1) The $\mod2$ reduction of $\vartheta$ is incomplete, which answers a question of Sarkaria.
(2) An odd-dimensional analog of $\vartheta$ is a complete obstruction to linkless embeddability ($=\,$“intrinsic unlinking”) of a given $n$-polyhedron in $\mathbb R^{2n+1}$.
(3) A “blown-up” one-parameter version of $\vartheta$ is a universal type 1 invariant of singular knots, i.e., knots in $\mathbb R^3$ 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 $\mathbb R^m$, $m\ge\frac{3(n+1)}2$, contains a subset $X$ such that no map $N\to\mathbb R^m$ sends $X$ and $N\setminus X$ 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 $\mathbb R^{2n-k}$ for $k<\frac{n-3}2$, thus extending the Penrose–Whitehead–Zeeman theorem.