  Given a finite CW complex $K$, we use a version of the Goodwillie-Weiss tower
to formulate an obstruction theory for embedding $K$ into a Euclidean space
$\mathbb{R}^d$. For $2$-dimensional 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. We focus on the first obstruction beyond the
classical embedding obstruction of van Kampen. In this case we show the two
approaches lead to essentially the same obstruction. We also give another
geometric interpretation of our obstruction, as a triple collinearity
condition. Furthermore, we relate our obstruction 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.
