On Sat, Nov 28, 2020 at 4:50 PM Allan Wechsler <acwacw@gmail.com> wrote:
The thing that is tripping me up is that the classes have to form a group, the first homotopy group of K. And the number of elements cited is prime, so the group would have to be cyclic.
They do not form a group, because of the requirement that these be *simple* closed curves, while the fundamental group looks at all maps from the circle to the space in question. For example, the fundamental group of the punctured plane is Z, but the number of inequivalent simple closed curves is 4; clockwise and counterclockwise circles that don't go around the removed point, and clockwise and counterclockwise circles that do go around the removed point. Any map of the circle to the punctured plane that goes twice around the missing point must self-intersect. Note that the requirement that the homotopy consist of simple closed curves means that the small clockwise and counterclockwise circle, both trivial as elements of the fundamental group, are not equivalent.
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Andy.Latto@pobox.com