The Circle is Not Simply Connected
In the comments to Number of Connected One-Dimensional Manifolds, I questioned why the circle (or more precisely the one-dimensional sphere S^1) was not simply connected. I wasn't trying to argue—I just didn't have the intuition myself, for some reason.
It's funny because now it's bleeding obvious to me that it isn't simply connected. A loop that goes from one point to another then back again clearly isn't homotopic to a loop that simply goes around the circle.
I think I was letting my intuition that S^n is simply connected override this fact. I was over generalising in my mind. S^n is simply connected only for n > 1. Thanks to Michael Hamm and Allan Engelhardt for setting me straight.
UPDATE: next post
Comments (1)
Created: Oct. 2, 2005
Last Modified: Aug. 9, 2007
Author: James Tauber
Last Modified: Aug. 9, 2007
Author: James Tauber
Adam Donahue on March 23, 2008:
Commenting on the math-based verification: this is something I've used myself for a couple years now, but rarely see used elsewhere. Pleasant surprise to see it used here, too. I tried using modern political questions (U.S. biased, admittedly), but was surprised at how many people can't name the Speaker of the House.