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

Moriya, Syunji; Sakai, Keiichi

doi:10.2140/agt.2018.18.2859

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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

DOI: 10.2140/agt.2018.18.2859

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 $ℝ^{3}$ . 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

Volume 18, issue 5 (2018)