On Sat, Nov 28, 2020 at 3:27 PM Dan Asimov <asimov@msri.org> wrote:
**********SPOILER********** for question 2. far below:
How many inequivalent simple closed curves are there on K, and what is an example of each one ???
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The Klein bottle has five different equivalence classes (of unoriented simple closed curves)
How many different equivalence classes of oriented simple closed curves are there? The fifth of your curve (the one that represents the trivial element of the fundamental group) is equivalent to its reverse, because the Klein bottle is nonorientable. Are any of the others equivalent to their reverses? Showing that the five curves you exhibit are inequivalent is straightforward, since they all correspond to different elements of the fundamental group. Can you give a hint as to how you would go about showing that these 5 are all that there are? Andy
. 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}
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