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. On Sat, Nov 28, 2020 at 3:27 PM Dan Asimov <asimov@msri.org> wrote:
**********SPOILER********** for question 2. far below:
On Wednesday/25November/2020, at 1:39 PM, Dan Asimov <asimov@msri.org> wrote:
Let K denote a Klein bottle.
1. The cartesian product
S^1 x S^2
of a circle (S^1) and a sphere (S^2) is a certain 3-dimensional manifold.
Puzzle: Does this manifold contain a Klein bottle
K ⊂ S^1 x S^2
as a subset ???
2. Call two simple closed curves C_0, C_1 on the Klein bottle "equivalent" if there is a continuous family
{C_t ⊂ K | 0 ≤ t ≤ 1}
of simple closed curves on K.
How many inequivalent simple closed curves are there on K, and what is an example of each one ???
|—————➞————| | | ↑ // ↓ | | |—————➞———— |
The Klein bottle has five different equivalence classes (of unoriented simple closed curves). Let the Klein bottle K be, as indicated by the picture, the result of identifying pairs of boundary points of the square [0,1]x[0,1] by
* (x,0) ~ (x,1), 0 ≤ x ≤ 1
and
* (0,y) ~ (1,1-y), 0 ≤ y ≤ 1.
Then the five classes are represented by these sets:
1) {(1/2,t) | 0 ≤ t ≤ 1}
2) {(t,0) | 0 ≤ t ≤ 1}
3) {(t,1/2) | 0 ≤ t ≤ 1}
4) {(t,1/4) | 0 ≤ t ≤ 1} ∪ {(t,3/4) | 0 ≤ t ≤ 1}
5) {(1/2+(1/4)cos(t), 1/2+(1/4)sin(t) | 0 ≤ t ≤ 2π}
—Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun