**********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