Embeddings of k-complexes into 2k-manifolds
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If K is a simplicial k-complex, the standard van Kampen obstructions tells you whether K can be embedded into R^2k or not (provided k≠ 2). We describe how the obstruction changes if we replace R^2k by a closed PL 2k-manifold satisfying a certain technical condition: We require that every map f |K| → M is homotopic to a map f' |K| → 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, Δ_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 ≥ 3, then the obstruction is complete, that is, we get the reverse implication. If M = S^2k (or 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 Z_2-reduction of the obstruction shows how to obtain a non-trivial upper bound for the Kühnel problem: determine the smallest n so that Δ_n^(k) does not embed into M. Also, the Z_2-reduction is computable and, if M is (k-1)-connected, it determines whether there is a map f |K| → M which has an even number of crossings of f(σ) and f(τ) for each pair (σ, τ) of disjoint k-simplices of K.
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