[Paper Review] Delooping of high-dimensional spaces of string links

Julien Ducoulombier|arXiv (Cornell University)|Sep 3, 2018

Algorithms and Data Compression13 references4 citations

TL;DR

This paper establishes explicit deloopings of high-dimensional spaces of string links modulo immersions using multivariable manifold calculus and derived mapping spaces of k-fold bimodules over operads. Under the condition $d_i + 3 eq n$ for all $i$, it proves that the homotopy fiber $L(d_1, dots, d_k; n)$ is homotopy equivalent to a $d$-fold iterated loop space, generalizing known results for $k=1$ to higher $k$.

ABSTRACT

We study a connection between a multivariable version of the Goodwillie-Weiss' calculus of functors and derived mapping spaces of k-fold bimodules over a family of operads. As our main application, under the assumption $d_{i}+3\leq n$ for all $i\in \{1,\ldots,k\}$, we produce explicit deloopings of high-dimensional spaces of string links modulo immersions from $\sqcup_{i}\mathbb{R}^{d_{i}}$ to $\mathbb{R}^{n}$ and their polynomial approximations.

Motivation & Objective

• To extend the known delooping of long knots modulo immersions ($k=1$) to the case of $k > 1$ string links in high dimensions.
• To understand the homotopy type of the space $\mathcal{L}(d_1,\ldots,d_k;n)$, the homotopy fiber of embedding spaces modulo immersions.
• To develop a multivariable version of Goodwillie-Weiss manifold calculus for $k$-fold infinitesimal bimodules and bimodules over operads.
• To construct explicit deloopings using derived mapping spaces in model categories of bimodules over families of operads.
• To generalize the rational homotopy description of $\mathcal{L}(d;n)$ to the multistrand case using graph complexes and coherent operads.

Proposed method

• Utilizes multivariable manifold calculus to analyze embedding spaces of $\coprod \mathbb{R}^{d_i}$ into $\mathbb{R}^n$.
• Applies the theory of $k$-fold infinitesimal bimodules and $k$-fold bimodules over operads to model embedding and immersion spaces.
• Constructs a Boardman-Vogt resolution for $k$-fold bimodules to resolve derived mapping spaces in the model category structure.
• Uses Reedy and projective model category structures on $k$-fold bimodules to define derived mapping spaces.
• Relies on the notion of coherent operads to relate the homotopy type of embedding spaces to iterated loop spaces.
• Applies the framework to the little cubes operads $\mathcal{C}_d$ and relates the results to graph complex homology via known equivalences.

Experimental results

Research questions

• RQ1 Can the delooping result for $\mathcal{L}(d;n)$, valid for $k=1$, be extended to $k > 1$ string links under suitable codimension conditions?
• RQ2 What is the homotopy type of the space $\mathcal{L}(d_1,\ldots,d_k;n)$, the homotopy fiber of embeddings modulo immersions?
• RQ3 How can multivariable manifold calculus be adapted to handle $k$-fold bimodules over families of operads?
• RQ4 What is the role of the Boardman-Vogt resolution in realizing derived mapping spaces for $k$-fold bimodules?
• RQ5 Can the rational homotopy type of $\mathcal{L}(d_1,\ldots,d_k;n)$ be described using graph complexes under weaker codimension assumptions than $2d_i+2\leq n$?

Key findings

• Under the condition $d_i + 3 \leq n$ for all $i$, the space $\mathcal{L}(d_1,\ldots,d_k;n)$ is homotopy equivalent to a $d$-fold iterated loop space, where $d = \min\{d_1,\ldots,d_k\}$.
• The homotopy fiber $\mathcal{L}(d_1,\ldots,d_k;n)$ is shown to be weakly equivalent to $\Omega^d \operatorname{Map}^{\text{h}}_{\text{Bimod}}(\mathcal{C}_d, \mathcal{C}_n)$, generalizing the $k=1$ case.
• The derived mapping space $\operatorname{Map}^{\text{h}}_{\text{Bimod}}(\mathcal{C}_d, \mathcal{C}_n)$ is realized as a $k$-fold bimodule over the little cubes operad $\mathcal{C}_d$.
• The paper constructs an explicit resolution of the $k$-fold infinitesimal bimodule $\mathbb{O}$ using the Boardman-Vogt construction in the $\Lambda^{\times k}$-setting.
• The model category structure on $k$-fold bimodules allows for the computation of derived mapping spaces, enabling the delooping construction.
• The results extend previous work on $k=1$ by providing a systematic framework for $k > 1$, with the codimension condition $d_i + 3 \leq n$ being optimal for the method.