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Given a finite CW complex $K$, we use a version of the Goodwillie-Weiss towerto formulate an obstruction theory for embedding $K$ into a Euclidean space$\mathbb{R}^d$. For $2$-dimensional complexes in $\mathbb{R}^4$, a geometricanalogue is also introduced, based on intersections of Whitney disks and moregenerally on the intersection theory of Whitney towers developed bySchneiderman and Teichner. We focus on the first obstruction beyond theclassical embedding obstruction of van Kampen. In this case we show the twoapproaches lead to essentially the same obstruction. We also give anothergeometric interpretation of our obstruction, as a triple collinearitycondition. Furthermore, we relate our obstruction to the Arnold class in thecohomology of configuration spaces. The obstructions are shown to be realizedin a family of examples. Conjectures are formulated, relating higher versionsof these homotopy-theoretic, geometric and cohomological theories.

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