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Feb 7, 2022 at 13:01 comment added Nick L For 1 I would be very surprised if it is not possible to perturb a path away from the $(n-2)$-skeleton (involving path lifting to the boundary map of a higher dimensional disk and then perturbing the lift), Then it seems we can arrange the condition by some local argument (involving path lifting to the boundary of the disk and then perturbing the lift). I am not 100% sure but can't really see how it would go wrong.
Feb 5, 2022 at 22:21 history asked Jake Levinson CC BY-SA 4.0