#### Volume 18, issue 5 (2018)

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

### Syunji Moriya and Keiichi Sakai

Algebraic & Geometric Topology 18 (2018) 2859–2873
##### Abstract

We prove affirmatively the conjecture raised by J Mostovoy (Topology 41 (2002) 435–450); the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in ${ℝ}^{3}\phantom{\rule{0.3em}{0ex}}$. We make use of techniques developed by S Galatius and O Randal-Williams (Geom. Topol. 14 (2010) 1243–1302) 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.

##### Keywords
topological category of long knots, the space of short ropes, classifying space, group completions, spaces of manifolds
##### Mathematical Subject Classification 2010
Primary: 57R19
Secondary: 55R35, 57M25