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I am given this quotient of the square:

enter image description here

I want to:

Find a homeomorphic CW complex,

Find its fundamental group,

Find a simply connected covering space

I have seen methods for other quotients of the square like the torus, and $\mathbb{R}P^2$ (for a presentation of the fundamental group), but they don't seem applicable here (in particular for the covering space).

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chilliBeanDream
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1 Answers1

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Hint: Cut-and-paste.

If you cut along the diagonal from upper left to lower right (marking the two cut-edges with, say, triple-arrows), and then glue together the two double-arrows, the resulting quotient description should be far easier to work with.

To answer your question literally, though: Lehrbuch der Topologie, by Seifert and Threlfall (also in English as "A textbook of topology"), has a nice description of how to do all this for a simplicial complex, although the presentation is rather dated. I think you'll find, if you read it, that in this case the distinction between "simplicial" and "CW" is compeletely irrelevant.

One more thing: If you know the Seifert/van Kampen theorem, computing the fundamental group is relatively easy: take as your set $U$ a concentric (open) square that's 3/4 as big in each direction. And take as $V$ everything outside a (closed) square that's 1/4 as big in each direction. Then $U \cap V$ is basically a circle, a circle that's contractible in $U$ and looks like $aa^{-1}bb$ in $V$, where $a$ and $b$ are the loops generated by the single-arrow and double arrow edges respectively. From this, you can compute $\pi_1(U \cup V)$.

John Hughes
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  • Thanks for the idea! I get something which is like a bag sealed along a line, so is that homeomorphic to a sphere? – chilliBeanDream Apr 03 '17 at 12:53
  • If that were the correct cut-and-paste result, then the answer would be "a sphere." But your cut-and-paste result is wrong. Take a square piece of paper, label the edges and diagonal with arrows (and then, for good measure, copy the labels to the back side of the paper), cut it along the diagonal, and reassemble so that the two double-arrows are aligned in the same direction (not "in opposite directions"!). See what you've got. (BTW, when you said "I get something which ...", I thought "Hunh...that sounds wrong..." and got out my pen and scissors and did just the thing I've just described.) – John Hughes Apr 03 '17 at 12:58
  • Oh yes, looks like I forgot to flip over one of them in my head, so its $\mathbb{R}P^2$ then. Thanks – chilliBeanDream Apr 03 '17 at 13:03
  • In response to the edit, I have a result about the fundamental group of gluing disks to spaces, from Hatcher. This gives a presentation $\langle a \mid a^2 \rangle$, which agrees with what I know about the space. – chilliBeanDream Apr 03 '17 at 13:13
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    Yep ... that's right. – John Hughes Apr 03 '17 at 13:14