We improve the bound on Kúhnel’s problem to determine the smallest n such that the k -skeleton of an n -simplex Δ n ( k ) does not embed into a compact PL 2 k -manifold M by showing that if Δ n ( k ) embeds into M , then n ≤ ( 2 k + 1 ) + ( k + 1 ) β k ( M ; 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 2 k -manifold M via the intersection form on M . In our approach we need that for every map f : K → 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 ≥ 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 2 k -manifolds.