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

TL;DR

This paper develops an obstruction theory for embedding finite CW complexes into Euclidean spaces using Goodwillie-Weiss calculus and Whitney disk intersection theory, connecting geometric, homotopy-theoretic, and cohomological perspectives.

Contribution

It introduces a unified obstruction framework for embeddings, linking Goodwillie-Weiss calculus with Whitney tower intersection theory and cohomological invariants.

Findings

First obstruction aligns with classical van Kampen obstruction

Obstruction can be interpreted as a triple collinearity condition

Obstructions are realized in explicit examples

Abstract

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.