The fundamental group considers only loops (not necessarily simple) starting and ending at a fixed choice of basepoint. This puzzle asks for only simple closed curves not necessarily starting & ending at any specific point. (The tiny local loop is the last one on the list of five classes I sent.) —Dan
On Saturday/28November/2020, at 1:49 PM, Allan Wechsler <acwacw@gmail.com> wrote:
I am very puzzled by the purported answer to the puzzle about loop classes on the Klein bottle. I don't see the "identity" class, the one whose exemplar is a tiny local loop. If that one is included, the answer would make sense to me.
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.