Twisting the unit square n times before gluing( 2.1.6 in G&P).

algebraic-topologydifferential-geometrydifferential-topologygeometric-topologymobius-band

The question is given below:

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I have made a Mobius band with a paper and twisted it 3-times but I could not describe what I see it may be a 3 knot shape, could anyone give me a hint for solving that question please?

EDIT:

The answer to my previous question is here:

Topologically distinguishing Mobius Strips based on the number of half-twists

But still I do not know how to prove that the boundary of the cylinder is 2 copies of $S^1$ and that of the Mobius band is 1 copy of $S^1$, could anyone help me in doing so?

Best Answer

Internally, you cannot tell a manifold with one even number of twists from one with some other even number. They are all orientable. The ones with an odd number of twists are non-orientable, but you can't tell one odd number from another. The boundary is $2$ copies of $S^1$ for the even case and $1$ copy of $S^1$ for the odd.

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