[Paper Review] On the homotopy type of the spaces of spherical knots in $R^n$

Victor Turchin, Thomas Willwacher|arXiv (Cornell University)|Dec 17, 2020

Homotopy and Cohomology in Algebraic Topology19 references4 citations

TL;DR

This paper establishes a homotopy equivalence between the space of spherical knots in ℝⁿ and the homotopy fiber of a map from the (n−m−1)-sphere to the classifying space of long knot spaces, revealing that the rational homotopy type of spherical knot spaces is determined by the hairy graph-complexes encoding long knot spaces. The key result shows that the space of spherical knots is a principal bundle over the sphere with structure group given by the long knot space, valid for codimension ≥3.

ABSTRACT

We study the spaces of embeddings $S^m\hookrightarrow R^n$ and those of long embeddings $R^m\hookrightarrow R^n$, i.e. embeddings of a fixed behavior outside a compact set. More precisely we look at the homotopy fiber of the inclusion of these spaces to the spaces of immersions. We find a natural fiber sequence relating these spaces. We also compare the $L_\infty$-algebras of diagrams that encode their rational homotopy type, when the codimension $n-m\geq 3$.

Motivation & Objective

• To understand the rational homotopy type of spaces of spherical knots in ℝⁿ.
• To relate the homotopy type of spherical knot spaces to that of long knot spaces via fiber sequences.
• To establish a homotopy equivalence between spherical knot spaces and a fiber bundle over the sphere with structure group given by long knot spaces.
• To express the rational homotopy type of spherical knot spaces through the same hairy graph-complexes used for long knot spaces.
• To extend known results on long knot spaces to spherical knots using homotopy-theoretic and operadic techniques.

Proposed method

• Use the homotopy fiber of the inclusion of embedding spaces into immersion spaces to define the spaces of interest: $\overline{\operatorname{Emb}}(S^m, \mathbb{R}^n)$ and $\overline{\operatorname{Emb}}_\partial(\mathbb{R}^m, \mathbb{R}^n)$.
• Construct a natural action of $\overline{\operatorname{Emb}}_\partial(\mathbb{R}^m, \mathbb{R}^n)$ on $\overline{\operatorname{Emb}}(S^m, \mathbb{R}^n)$ via perturbation near a point on the sphere.
• Prove that the homotopy quotient of $\overline{\operatorname{Emb}}(S^m, \mathbb{R}^n)$ by this action is homotopy equivalent to $S^{n-m-1}$, leading to the fiber sequence in Theorem 1.1.
• Use rational homotopy theory and $L_\infty$-algebras to compare the rational homotopy types of the two spaces.
• Employ hairy graph-complexes $\mathrm{HGC}_{\bar{A}_{m},n}$ to compute the rational homotopy groups of $\overline{\operatorname{Emb}}_\partial(\mathbb{R}^m, \mathbb{R}^n)$, and extend this to spherical knots via the main equivalence.
• Apply the generalized Goldman-Millson theorem to show that the Maurer-Cartan spaces of the relevant $L_\infty$-algebras are equivalent, implying isomorphism of rational homotopy types.

Experimental results

Research questions

• RQ1 How are the rational homotopy types of spaces of spherical knots in ℝⁿ related to those of long knots in ℝⁿ?
• RQ2 Can the homotopy fiber of the inclusion $\operatorname{Emb}(S^m, \mathbb{R}^n) \to \operatorname{Imm}(S^m, \mathbb{R}^n)$ be described in terms of a fiber bundle over a sphere?
• RQ3 What is the role of the long knot space $\overline{\operatorname{Emb}}_\partial(\mathbb{R}^m, \mathbb{R}^n)$ in controlling the homotopy type of the spherical knot space?
• RQ4 In codimension $n-m=2$, can the rational homotopy type of spherical knot spaces still be expressed via graph-complexes despite infinite series of graphs?
• RQ5 How does the $L_\infty$-algebra structure of the hairy graph-complex relate to the rational homotopy type of the spherical knot space?

Key findings

• The space $\overline{\operatorname{Emb}}(S^m, \mathbb{R}^n)$ is homotopy equivalent to the homotopy fiber of a map $S^{n-m-1} \to B\overline{\operatorname{Emb}}_\partial(\mathbb{R}^m, \mathbb{R}^n)$, establishing a principal bundle structure.
• All connected components of $\overline{\operatorname{Emb}}(S^m, \mathbb{R}^n)$ have the same rational homotopy type, and $\pi_0$ of the space agrees with that of the long knot space.
• For $n-m \geq 3$, the rational homotopy type of $\overline{\operatorname{Emb}}(S^m, \mathbb{R}^n)^\mathbb{Q}$ is equivalent to $\mathsf{MC}_\bullet(\mathrm{HGC}_{A_{m},n})$, which computes the rational homotopy groups via the hairy graph-complex.
• In codimension $n-m=2$, the rational homotopy type is still expressible via completed $L_\infty$-algebras, with $T_\infty\overline{\operatorname{Emb}}(S^m, \mathbb{R}^n)^\mathbb{Q} \simeq K(\mathbb{Q},1) \times T_\infty\overline{\operatorname{Emb}}_\partial(\mathbb{R}^{n-2}, \mathbb{R}^n)^\mathbb{Q}$ for $n=3,5,9$, due to the presence of a one-dimensional $L_\infty$-subalgebra.
• The rational homotopy types of $\overline{\operatorname{Emb}}(S^m, \mathbb{R}^n)^\mathbb{Q}$ and $\overline{\operatorname{Emb}}_\partial(\mathbb{R}^m, \mathbb{R}^n)^\mathbb{Q}$ are isomorphic in the range $n-m \geq 3$ or $n=m+2=3,5,9$, via the equivalence $\mathsf{MC}_\bullet(U^t \oplus \mathrm{HGC}_{\bar{A}_{m},n}) \simeq \mathsf{MC}_\bullet(\mathrm{HGC}_{A_{m},n})$.