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Feb 4, 2010 at 1:00 comment added Ryan Budney This is in spirit very closely related to my underlying motivation -- spaces of knots have a natural space-level group completion as they can be turned into topological monoids by considering the corresponding "long knot" space. At present I'm not sure what the group completion should be other than the knowledge that it has a certain iterated loop-space structure so I'm hoping that knowledge of $\pi_0$ would give some insights.
Feb 3, 2010 at 22:36 history answered Oscar Randal-Williams CC BY-SA 2.5