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

The original post was in the category: poincare_project but I'm still in the process of migrating categories over.

The original post had 1 comment I'm in the process of migrating over.