  We improve the bound on K\"uhnel's problem to determine the smallest $n$ such
that the $k$-skeleton of an $n$-simplex $\Delta_n^{(k)}$ does not embed into a
compact PL $2k$-manifold $M$ by showing that if $\Delta_n^{(k)}$ embeds into
$M$, then $n\leq (2k+1)+(k+1)\beta_k(M;\mathbb Z_2)$. 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 $f\colon K\to M$ 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 $k\geq 3$, 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.
