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

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.