Asian Journal of Mathematics

Volume 27 (2023)

Number 2

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

Pages: 135 – 186

DOI: https://dx.doi.org/10.4310/AJM.2023.v27.n2.a1

Authors

Gregory Arone (Department of Mathematics, Stockholm University, Stockholm, Sweden)

Vyacheslav Krushkal (Department of Mathematics, University of Virginia, Charlottesville, Va., U.S.A.)

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.

Keywords

embeddings, configuration spaces, Goodwillie–Weiss calculus, Whitney disks

2010 Mathematics Subject Classification

55P65, 55R80, 57N35, 57Q35

Received 22 July 2021

Accepted 26 January 2023

Published 12 October 2023