10

We define antipodal points in $\mathbb{R}^3$ as $(x,y,z)\to (-x,-y,-z)$. As we all know, the 2-torus can be expressed as $$ \left(\sqrt{x^2+y^2}-R\right)^2+z^2=r^2, $$ where $0<r<R$.

Some were saying it was a Klein bottle. So I am considering that since torus can be regarded as a square with opposite edges identified in the same order. By identifying the antipodal points, we should also identify the interior of the identified square, so it is not quite the same as we have for Klein bottle. Can anyone hint on this? Thanks!

KasparJohannes
  • 161
jack
  • 525
  • 1
    If you think of Torus as gotten from Square by identifying boundary points in same order, what is the antipode of a given point? – Lubin Jan 12 '17 at 21:05
  • @Lubin I am new to this... Could you please elaborate on this? It would be greatly appreciated if you could show some graph... Thanks. – jack Jan 12 '17 at 21:10
  • Hint: under this identification, any point is an image of a point with $z \ge 0$. The projection of the part of your $2$-torus with $z \ge 0$ onto the plane $z = 0$ shows that it is homeomorphic with an annulus. Draw how the antipodal identification works on the boundary circles of the annulus and then cut the annulus along a radial line to get a rectangle with identifications around the boundary. – Rob Arthan Jan 12 '17 at 21:11
  • In your definition intersection points obtained by cutting by all lines through origin are anti-podal? – Narasimham Jan 12 '17 at 21:11
  • Well, the antipodal map would be a transformation of your square which gives the identity when composed with itself, in other words, your antipodal map would be its own inverse. I myself don’t know a good choice for such a map that would obviously suffice for your purpose. – Lubin Jan 12 '17 at 21:14

1 Answers1

9

This identification yields a Klein bottle. In particular, you can think of slicing the torus by a half-space passing through the origin, then identifying the circles cut by the boundary of this plane in an appropriate way, and seeing that the two circles one is supposed to identify are oriented in opposite directions.

More formally, let us define a torus by coordinates $(\theta,\psi)$ sending a pair of these to the coordinates $$(R\sin(\theta)+r\cos(\psi)\sin(\theta),R\cos(\theta)+r\cos(\psi)\cos(\theta),r\sin(\psi)).$$ One can observe that your identification identifies pairs $(\theta_1,\psi_1)$ and $(\theta_2,\psi_2)$ where $\theta_2-\theta_1=\pi$ and $\psi_1=-\psi_2$, taking these equations mod $2\pi$. In particular, one can quickly see that, every point in this quotient space has a unique representative with $\theta\in [0,\pi)$ and $\psi\in [0,2\pi)$ and we can see that points of the form $(\pi,\psi)$ correspond to ones of the form $(\pi,-\psi)$ and ones of the form $(\theta,2\pi)$ correspond to $(\theta,0)$. In particular, one sees that this space is the square $[0,\pi]\times [0,2\pi]$ achieved by identifying the top and bottom edges in the same direction, and the left and right edges in opposite directions, which is a Klein bottle.

Milo Brandt
  • 60,888