If K is a simplicial k-complex , the standard van Kampen obstructions tells you whether K can be embedded into \mathbb R^{2k or not (provided k\neq 2). We describe how the obstruction changes if we replace \mathbb R^{2k} by a closed PL 2k-manifold satisfying a certain technical condition : We require }  that every map f\colon |K| \to M is homotopic to a map f'\colon |K| \to M such that f'(|K^{(k-1)}|) fits into some 2k-ball in M, where K^{(k-1)} stands for the (k-1)-skeleton of K. The technical condition is satisfied, in particular, either if M is (k-1)-connected or if K is the k-skeleton of n-simplex, \Delta_n^{(k)}, for some n. Under the technical condition, if K embeds in M, then our obstruction vanishes. In addition, if M is (k-1)-connected and k \geq 3, then the obstruction is complete, that is, we get the reverse implication. If M = S^{2k} (or \mathbb R ^{2k}) then the intersection form on M vanishes and our obstruction coincides with the standard van Kampen obstruction. However, if the intersection form is nontrivial, then our obstruction is not linear (a cohomology class) but rather `quadratic' in a sense that it vanishes if and only if a certain system of quadratic equations over integers is solvable. It remains to be determined whether these systems can be solved algorithmically. Finally, the \mathbb Z _2-reduction of the obstruction shows how to obtain a non-trivial upper bound for the K\"{u}hnel problem: determine the smallest n so that \Delta_n^{(k)} does not embed into M. Also, the \mathbb Z _2-reduction is computable and, if M is (k-1)-connected, it determines whether there is a map f\colon |K| \to M which has an even number of crossings of f(\sigma) and f(\tau) for each pair (\sigma, \tau) of disjoint k-simplices of K.
Let K be a simplicial k-complex or not (provided k\neq 2). We describe how the obstruction changes if we replace \mathbb R^{2k} by a closed PL 2k-manifold satisfying a certain technical condition : We require } and M be a closed PL 2k-manifold. Our aim is to describe an obstruction for embeddability of K into M via the intersection form on M. For description of the obstruction, we need a technical condition that every map f\colon |K| \to M is homotopic to a map f'\colon |K| \to M such that f'(|K^{(k-1)}|) fits into some 2k-ball in M, where K^{(k-1)} stands for the (k-1)-skeleton of K. The technical condition is satisfied, in particular, either if M is (k-1)-connected or if K is the k-skeleton of n-simplex, \Delta_n^{(k)}, for some n. Under the technical condition, if K embeds in M, then our obstruction vanishes. In addition, if M is (k-1)-connected and k \geq 3, then the obstruction is complete, that is, we get the reverse implication. If M = S^{2k} (or \mathbb{R ^{2k}) then the intersection form on M vanishes and our obstruction coincides with the standard van Kampen obstruction. However, if the intersection form is nontrivial, then our obstruction is not linear (a cohomology class) but rather `quadratic' in a sense that it vanishes if and only if a certain system of quadratic equations over integers is solvable. It remains to be determined whether these systems can be solved algorithmically. Finally, the \mathbb{Z _2-reduction of the obstruction shows how to obtain a non-trivial upper bound for the K\"{u}hnel problem: determine the smallest n so that \Delta_n^{(k)} does not embed into M. Also, the \mathbb{Z _2-reduction is computable and, if M is (k-1)-connected, it determines whether there is a map f\colon |K| \to M which has an even number of crossings of f(\sigma) and f(\tau) for each pair (\sigma, \tau) of disjoint k-simplices of K.
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and M be a closed PL 2k-manifold. Our aim is to describe an obstruction for embeddability of K into M via the intersection form on M. For description of the obstruction, we need a technical condition
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