  The vanishing of Van Kampen's obstruction is known to be necessary and
sufficient for embeddability of a simplicial n-complex into $R^{2n}$ for $n\neq
2$, and it was recently shown to be incomplete for $n=2$. We use
algebraic-topological invariants of four-manifolds with boundary to introduce a
sequence of higher embedding obstructions for a class of 2-complexes in $R^4$.
