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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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 3. 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
References
Publication
Received: 29 May 2017
Revised: 24 January 2018
Accepted: 22 February 2018
Published: 22 August 2018
Authors
Syunji Moriya
Department of Mathematics and Information Sciences
Osaka Prefecture University
Sakai
Japan
Keiichi Sakai
Department of Mathematical Sciences
Shinshu University
Asahi
Matsumoto
Japan
http://math.shinshu-u.ac.jp/~ksakai/index.html