Embedding obstructions and 4-dimensional thickenings of 2-complexes

The vanishing of Van Kampen’s obstruction is known to be necessary and sufficient for embeddability of a simplicial n n -complex into R 2 n {\mathbb {R}}^{2n} for n ≠ 2 n\neq 2 , and it was recently shown to be incomplete for n = 2 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 2 -complexes in R 4 {\mathbb {R}}^4 .