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The Möbius band has the polygonal presentation $$\langle a,b,c \vert abcb\rangle$$

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Now I am asked to identify the boundary of the Möbius band to a single point. How do I represent this using polygonal presentations? Is that even possible?

I am aware of the same question here and my intuition was also $\mathbb{P}^2$ but I want to prove this using presentations rather than just words. I mean it is a general question, how to present the identification to a single point.

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TheGeekGreek
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1 Answers1

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In your square diagram, pinch sides $a$ and $c$ to points, leaving the "digon" whose sides are the $b$'s with indicated identification. That's a well-known representation of the real projective plane.

Andrew D. Hwang
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    Nice, thank you. I wondered if there are any intermediate steps, but apparently not. Thanks. – TheGeekGreek Oct 30 '16 at 12:52
  • Instead of shrinking the boundary to a point you can add a 2-cell (you can't do this in 3-space.) The following gives some relevant pictures:http://www.groupoids.org.uk/outofline/motion.html#motion – Ronnie Brown Oct 30 '16 at 15:27