  Given an $m$-dimensional CW complex $K$, we use a version of the
Goodwillie-Weiss tower to formulate an obstruction theory for embeddings into a
Euclidean space ${\mathbb R}^d$. For $2$-complexes in ${\mathbb R}^4$ a
geometric analogue is also introduced, based on intersections of Whitney disks
and more generally on the intersection theory of Whitney towers developed by
Schneiderman and Teichner. The focus in this paper is on the first obstruction
beyond the classical embedding obstruction of van Kampen. In this case we show
the two approaches give the same result, and also relate it to the Arnold class
in the cohomology of configuration spaces. The obstructions are shown to be
realized in a family of examples. Conjectures are formulated, relating higher
versions of these homotopy-theoretic, geometric and cohomological theories.
