The van Kampen Obstruction and Its Relatives

S. A. Melikhov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometry, topology, and mathematical physics. II, Tome 266 (2009), pp. 149-183.

Résumé

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.

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