Handles, Crosshandles, Crosscaps and Co

Question

are there some fans of funny and challenging topology exercises?! For now i proved the following:

A Klein Bottle is homeomorphic to the union of two Möbiusstrips along their boundaries
A crosshandle (or twisted handle) is homeomorphic to two crosscaps

For the first proof there is no problem (in addition there is a proof given here "Klein-bottle and Möbius-strip together with a homeomorphism" but you can do this proof on many different ways). For the second proof i supposed $X$ to be the sphere with a crosshandle and $Y$ to be the sphere with two crosscaps. Then i made a homeomorphism $\phi:X\rightarrow Y$ (because you can give arguments that both spaces are Klein-bottles). Is there someone who can give good arguments herefore? (i also will give my own arguments if someone is asking).

But now i want to prove that the a crosscap with a handle is homeomorphic to three crosscaps (or in other words: i want to give a proof of Dyck's Theorem (true?!) Can i use the facts above?? Can someone help me with this construction?

Dave · Accepted Answer · 2013-07-14T12:56:11.250

3

You might be interested in the paper Conway's ZIP Proof by G. Francis & J. Weeks. If I remember correctly, everything you're looking for is nicely described in there.

edited Jul 14 '13 at 12:56

answered Feb 27 '13 at 12:29

Dave

1,853

Okay thanks :) Maybe it is the best idea to read the script and figure out what is important for the exercise. – Infinity Feb 27 '13 at 12:33
Link doesn't seem to work anymore. – Ben Millwood Jul 14 '13 at 01:01
@BenMillwood : Thank you, I fixed the link. – Dave Jul 14 '13 at 12:57

Handles, Crosshandles, Crosscaps and Co

1 Answers1