An obstruction to embedding a simplicial n-complex into a 2n-manifold

Let K be a finite, connected, simplicial n-complex (n⩾3) and M a 1-connected, smooth, orientable 2n-manifold without boundary. If f :|K|→M is an arbitrary map, we define a first obstruction γ(f)∈H2n(J∗K;Z), where J∗K is the reduced deleted product of K and show that the vanishing of this obstruction is necessary and sufficient for f to be homotopic to an embedding.