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Question 1: How exactly a crosscap relates to a Mobius strip, a $^2$, and a Klein bottle?

I knew that

(1) Mobius strip glued the boundary with a disk $D^2$ is $^2$.

(2) Gluing the boundaries of two Mobius strip gives a Klein bottle

(3) Connecting sum two $^2$ gives $^2 \# ^2 $ = Klein bottle.

Euler characteristic $\chi$:

  • Mobius strip with $\chi=0$

  • Disk with $\chi=1$

  • $^2$ with $\chi=1$

  • Klein bottle with $\chi=0$

  • Question 2: Crosscap has $\chi=?$

annie marie cœur
  • 3,857
  • Crosscap is just another name for the Moebius band. – Moishe Kohan Nov 25 '20 at 00:09
  • But Moebius band has 1 boundary, but the Crosscap has no boundary. Can we glue the boundary of a Moebius band in some way to get a Crosscap? But a Moebius band glued the boundary with a disk 2 is 2...please illuminate? – annie marie cœur Nov 25 '20 at 01:00
  • It depends on what you mean by the crosscap. What is your definition? – Moishe Kohan Nov 25 '20 at 01:02
  • definition from Wiki picture? – annie marie cœur Nov 25 '20 at 01:03
  • Sadly, what's written there is not a mathematical definition. They are attempting to describe a map from $RP^2$ to $R^3$. When one says "add a cross-cup to a surface with boundary", they mean "glue the Moebius band along a boundary curve." – Moishe Kohan Nov 25 '20 at 01:05
  • My general suggestion is to read a topology textbook instead of relying on Wikipedia. – Moishe Kohan Nov 25 '20 at 01:07
  • which topology books (as many as good as possible) you suggest? Lol – annie marie cœur Nov 25 '20 at 01:20
  • My favorite for classification of surfaces in Massey's "Algebraic Topology." – Moishe Kohan Nov 25 '20 at 01:23
  • Thanks, how about I use the definition in the Fig 1 of this paper? – annie marie cœur Nov 25 '20 at 01:33
  • They are so sloppy, it's hard to tell what they mean: When gluing surfaces from polygons one usually draws arrows to indicate the orientation, but they also use circles which a topologist would never use in this setting. For instance in part a of figure 1 it is either projective plane or the Klein bottle, depending on the orientation of the edges labeled by blue circles. In part d, you either get the torus or the Klein bottle. As I suggested, read Massey's book. – Moishe Kohan Nov 25 '20 at 01:43

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