Is a crosshandle homeomorphic to a Klein bottle?

Question

I am aware that a Klein bottle is homeomorphic to two Möbius bands, and by Conway's zip proof a crosshandle is homeomorphic to two crosscaps.

Now, since you can think of a crosscap as a Möbius band sewn into a circular hole in a surface, I am wondering if this means a crosshandle is homeomorphic to a Klein bottle.

Intuitively this would make sense because in the construction of the crosshandle we see a surface "pass through itself" in a similar way to the Klein bottle but I am struggling to explain this rigorously.

Thanks!

Answer 1

Yes.

Every non-orientable surface is homeomorphic to the connected sum of some number of crosscaps, the number of which is determined by the surface’s Euler characteristic. The Klein bottle has Euler characteristic zero, as does the connected sum of two crosscaps. Thus, they are homeomorphic.