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.
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.