Embeddings of $k$-complexes into $2k$-manifolds

We improve the bound on K"uhnel’s problem to determine the smallest $n$ such that the $k$-skeleton of an $n$-simplex ${\mathrm{\Delta }}_n^{(k)}$ does not embed into a compact PL $2k$-manifold $M$ by showing that if ${\mathrm{\Delta }}_n^{(k)}$ embeds into $M$, then $n\le (2k+1)+(k+1){\beta }_k(M;{\mathbb{Z}}_2)$. As a consequence we obtain improved Radon and Helly type results for set systems in such manifolds. Our main tool is a new description of an obstruction for embeddability of a $k$-complex $K$ into a compact PL $2k$-manifold $M$ via the intersection form on $M$. In our approach we need that for every map $f\text{\hspace{0.17em}⁣}:K\to M$ the restriction to the $(k-1)$-skeleton of $K$ is nullhomotopic. In particular, this condition is satisfied in interesting cases if $K$ is $(k-1)$-connected, for example a $k$-skeleton of $n$-simplex, or if $M$ is $(k-1)$-connected. In addition, if $M$ is $(k-1)$-connected and $k\ge 3$, the obstruction is complete, meaning that a $k$-complex $K$ embeds into $M$ if and only if the obstruction vanishes. For trivial intersection forms, our obstruction coincides with the standard van Kampen obstruction. However, if the form is non-trivial, the obstruction is not linear but rather ‘quadratic’ in a sense that it vanishes if and only if certain system of quadratic diophantine equations is solvable. This may potentially be useful in attacking algorithmic decidability of embeddability of $k$-complexes into PL $2k$-manifolds.