1. ## Klein Bottle Covering

Consider the equivalence relation ~ on $\displaystyle \mathbb{R}^2$.
$\displaystyle (x,y) \sim (x,y+1)$ and $\displaystyle (x,y) \sim (x+1,-y)$

The quotient $\displaystyle K = \mathbb{R}^2/ \sim$ is the Klein Bottle. Show that the quotient map $\displaystyle p: \mathbb{R}^2\rightarrow K$ is a covering.

Dont really know how to start I know I need to find an open cover of $\displaystyle U_i$ wherethe preimage of each $\displaystyle U_j$ under p is a disjoint union of open sets of which each is homeomorphic to $\displaystyle U_j$.
Any hints are welcome. I basically dont see how this equivalence relation works. I know the Klein Bottle as quotient of I^2 with (0,y) ~ (1,y) and (x,0) ~ (1-x,1). Is this somehow related to this task?

2. ## Re: Klein Bottle Covering

choose any point, for example a=p(0,0), on K. choose a small open disk D centered at it with radius r<1/2,
Then the pre-image of D is the union of disks centered at pre-images of a, which are all the points with integer coordinates.

3. ## Re: Klein Bottle Covering

Ok, you use the fact that all points equivalent to (0,0) are exactly the points (m,n) with m and n being integers. But what happens if you choose a non integer points, I mean where is the difference for example to the quotient map of the torus (i.e. where do we use the fact that the equivalence relation is not just (x,y) ~ (x,y+1) and (x,y) ~ (x+1,y) ? )

4. ## Re: Klein Bottle Covering

If you choose a non-integer point (x0,y0), then the equivalent points are (x0+m,y0+n) where m,n are integers.
Locally there is no difference. But if you travel along a horizonal line in the plane, starting from (x,y), you jumps to different points after you proceed 1 unit of distance,
That is, for a torus, from (x+1,y) to (x,y)
for a Klein bottle, from (x+1,y) to (x,-y)

5. ## Re: Klein Bottle Covering

suppose we consider just the vertical strip IxR, and take a look at an open disk U of say, radius 1/8 on K, at the point p(1,1/2).

then p^-1(U)∩(IxR) consists of a (left) half-circle of radius 1/8 centered at (1,1/2) and a (right) half-circle centered at (0,-1/2), plus vertical translates of these half-circles at unit intervals along IxR.

if we extend our strip, to all of RxR, we see that we have alternating half-circles every unit (on the y-axis we have the "right half-circles", on x = 1, we have the "left haf-circles, etc.) horizontally, which also extend vertically.

only disks which cross p({n}xR) display this unusual behavior (this is due to the "twist" of the klein bottle, whereas disks that intersect p(Rx{n}) just "unwrap" cylindrically, and their pre-images are just circles spaced at unit intervals).

6. ## Re: Klein Bottle Covering

So to prove that the quotient map p is a covering, I need to find an open cover of K, which open cover would be suitable?