Embedding obstructions in ${\mathbb R}^d$ from the Goodwillie-Weiss calculus and Whitney disks

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.