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

Gregory Arone, Vyacheslav Krushkal

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.

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

Abstract

Given a finite CW complex , we use a version of the Goodwillie-Weiss tower to formulate an obstruction theory for embedding into a Euclidean space . For -dimensional complexes in , 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.

Paper Structure

This paper contains 27 sections, 28 theorems, 118 equations, 15 figures.

Table of Contents

  1. Introduction
  2. The van Kampen obstruction from ${\mathbf 2}$-point configuration spaces
  3. The obstruction from ${\mathbf 3}$-point configuration spaces
  4. ${\mathbf 2}$-complexes in ${\mathbf \mathbb{R}^4}$: obstructions from intersections of Whitney disks
  5. Lift of the obstructions from cohomology to framed cobordism
  6. Outline of the paper
  7. The first and second cohomological obstructions to embedding
  8. The van Kampen obstruction
  9. The secondary obstruction
  10. A lift to framed cobordism
  11. Geometric obstructions from Whitney towers
  12. The van Kampen obstruction
  13. Operations on Whitney disks
  14. From Whitney disks to equivariant maps of configuration spaces
  15. An obstruction from intersections of Whitney disks
  16. ...and 12 more sections

Key Result

Theorem 1.3

The homotopy-theoretic obstruction $\operatorname{{\mathcal{O}}_2\!}{(K)}$ agrees with the geometric obstruction $\operatorname{{\mathcal{W}}_2\!}{(K)}$. When $2\dim(K)=d$, $\operatorname{{\mathcal{O}}_2\!}{(K)}$ is a complete obstruction for $T_2\operatorname{Emb}(K, \mathbb{R}^d)$ to be non-empty.

Figures (15)

  • Figure 1: Finger move: homotopy of maps $f\colon K\longrightarrow {\mathbb R}^{2m}$
  • Figure 2: A Whitney disk and the associated capped surface
  • Figure 3: The result of the Whitney move
  • Figure 4: Left: the Borromean rings in $\partial D^4$. Right: The Whitney disk $W_{ij}$ intersects $\sigma_k$ in a single point
  • Figure 5: Splitting of a Whitney disk
  • ...and 10 more figures

Theorems & Definitions (69)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 59 more