Before revision:  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.
After revision:  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.