The space of short ropes and the classifying space of the space of long knots

Abstract

We prove affirmatively the conjecture raised by J. Mostovoy [3]; the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in R 3 . We make use of techniques developed by S. Galatius and O. Randal-Williams [2] to construct a manifold space model of the classifying space of the space of long knots, and we give an explicit map from the space of short ropes to the model in a geometric way.

Delooping the functor calculus tower

“…For other approaches of delooping the spaces of disc embeddings and related problems, see also [13,48,49,51].…”

“…It has recently been shown by Willwacher and the authors in [20] that $$\begin{equation*} {\mathrm{Emb}}_\partial ^{fr}(D^m,D^n)\simeq \Omega ^{m+1}{\left(\, {\mathrm{Operad}}^h({\mathcal {B}}_m,{\mathcal {B}}_n)\sslash SO(n)\,\right)}, \,\, n\geqslant m+3; \end{equation*}$$$$\begin{equation*} T_k {\mathrm{Emb}}_\partial ^{fr}(D^m,D^n)\simeq \Omega ^{m+1}{\left(\, {\mathrm{T}}_k{\mathrm{Operad}}^h({\mathrm{T}}_k{\mathcal {B}}_m,{\mathrm{T}}_k {\mathcal {B}}_n)\sslash SO(n)\,\right)}. \end{equation*}$$For other approaches of delooping the spaces of disc embeddings and related problems, see also [13, 48, 49, 51].…”

On the delooping of (framed) embedding spaces

“…Given an integer k ≥ 1, we also consider the category of k-truncated reduced bimodules over O denoted by Bimod O ; ≤k . The objects are k-truncated Λ-sequences together with operations of the form (13) with n + m − 1 ≤ k for the right operations and m 1 + • • • + m n ≤ k for the left operation. Furthermore, one has the functor…”

“…, n 4. For other related results on the little discs action on the spaces of disc embeddings and results on their deloopings we refer the reader to [1,4,5,8,11,[13][14][15][16][17].…”