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

概要

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 R3. 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.