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I am looking at a past exam paper for my introductory algebraic topology course, and am asked, for each of the following identification spaces, to find a CW complex homeomorphic to the space, draw the corresponding 1-skeleton, give a presentation of its fundamental group, and find its universal cover.

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I am primarily having trouble with the last part, since I have no examples of universal covers in my notes, only a long proof that every path connected pointed CW complex $(X,x_0)$ has a path-connected pointed universal cover $(\tilde{X},\tilde{x}_0)$, which is unique up to isomorphism. I will explain what I have done so far - in each case I am finding the presentation by starting in the bottom left corner and traversing the sides anti-clockwise, with $a$ the generator corresponding to the single arrowed sides, and $b$ the generator corresponding to the double arrowed sides. Considering each space from left to right:

i) Presentation is $G=\langle{a,b|aba^{-1}b^{-1}}\rangle$, CW complex structure is one 0-cell, two 1-cells (both loops based at the 0-cell), and one 2-cell, whose boundary goes around $a,b$, then $a$ reversed and $b$ reversed. The 1-skeleton is then $\mathbb{S}^1\vee\mathbb{S}^1$.

ii) Presentation is $G=\langle{a,b|aa^{-1}bb}\rangle=\langle{a,b|b^2}\rangle$. I can't really get my head around the CW structure - the bottom left, top left and right corners are identified into one 0-cell, and the bottom right is another 0-cell?

iii) Presentation is $G=\langle{a,b|aa^{-1}bb^{-1}}\rangle=\langle{a,b|~}\rangle$. I'm imagining this CW structure to be a sphere, with three 0-cells joined together by two 1-cells? I'm not sure how to describe the attachment of the 2-cell. The 1-skeleton would just be the three 0-cells joined by two 1-cells; I'll try to illustrate this as: \begin{equation} \cdot-\cdot-\cdot \end{equation}

In any case I have no idea where to start in finding universal covers. I'm also quite unsure of how exactly I should be describing the attachment of the 2-cells when describing the CW structure on each of these. Any help would be appreciated.

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  • $\begingroup$ Your presentations for II and iii are not correct. $\endgroup$ – user98602 Apr 5 '16 at 15:42
  • $\begingroup$ A quick comment for ii): One of the edges is a loop, but the other is not, so does not give an element in $\pi_1$. $\endgroup$ – Cheerful Parsnip Apr 5 '16 at 15:42
  • $\begingroup$ @MikeMiller could you please explain why not? $\endgroup$ – jl2 Apr 5 '16 at 15:53
  • $\begingroup$ "Because they give different groups than the correct ones." I think it's a better learning experience to see yourself what goes wrong! $\endgroup$ – user98602 Apr 5 '16 at 15:54
  • $\begingroup$ @GrumpyParsnip is that to say that $a$ doesn't belong in the presentation since it doesn't generate a loop? $\endgroup$ – jl2 Apr 5 '16 at 15:57

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