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I am reading the first chapter from Topology by Armstrong. There, after stating the classification theorem for closed surfaces, he has mentioned an example that a sphere with one handle and one Möbius strip glued is homeomorphic to a sphere with three Möbius strips glued.

I am not able to see this. I know that to prove the above is to say that a torus with a Möbius strip glued is homeomorphic to a Klein bottle with a Möbius strip glued. How do I prove this? Or how do I at least convince myself that this is the case, in case I don't yet have the tools to prove it.

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Are you familiar with the Euler characteristic of a surface and how it changes when taking connected sums? – Miha Habič Jan 5 '13 at 8:16
No I dont know much about it. Is there not a direct approach? – user52991 Jan 5 '13 at 13:51
Polygonal models of surfaces should be enough to prove this. The torus is a square with opposite sides identified in natural way. The Moebius band is a square with a pair of opposite sides identified in an unnatural way. Connected sums can be performed by chopping off a vertex on each diagram (leaving a short edge in its place) and identifing these new edges. Then some cutting and pasting may be necessary to bring the two connected sums to the same polygonal form. – user53153 Jan 6 '13 at 2:40

See the proof of Lemma 7.1 in Massey's "A Basic Course in Algebraic Topology" (1991), pp 23-25.

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Could you summarize the basic idea of the proof here? It would be helpful for people without immediate access to the textbook. – Rahul Jan 6 '13 at 18:40

Mobius strip allows access into the inside surface of the sphere. So slide one end of the g-handle into the strip and and it will then become attached to the sphere from the inside surface. That effectively turns the g-handle into a klien bottle and every such bottle is made up of two mobius strips.

So there, the three mobius strips you mentioned.

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