Embeddings of k-complexes into 2k-manifolds
Abstract
We improve the bound on K\"uhnel's problem to determine the smallest n such that the k-skeleton of an n-simplex n(k) does not embed into a compact PL 2k-manifold M by showing that if n(k) embeds into M, then n≤ (2k+1)+(k+1)βk(M; Z2). 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 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 2k-manifolds.