I'm studying for an exam, and got stuck on the following exercise:

Find all two-sheeted covering spaces for $X =\mathbb{S}^1 \vee \mathbb{S}^1$.

Label the two circles of $X$ by $a$ and $b$. Attach a two-cell to $X$ with an attaching map $aba^{-1}$ to create a new space $Y$. Which of the coverings found above are restrictions of covering spaces of $Y$?

For the coverings of $X$, it's fairly simple to see that there are three connected two-sheeted coverings (they're drawn, for instance, here). The last two-sheeted covering is the one that simply consists of two disjoint copies of $X$ which project down.

I don't understand how to check whether the coverings are restrictions of covers for $Y$ though. I can see, of course, that the non-connected cover is a restriction of the non-connected two-sheeted cover for $Y$, but for the others, I'm not sure what to do.

Any ideas? Thanks.


1 Answer 1


The interior of the disk attached by $aba^{-1}$ must lift to two copies of that disk since the only covers of a simply connected space are trivial or disconnected. If you look at the boundary of the disk you can glue up the $a$ and $a^{-1}$ to get a new disk which attaches around its perimeter by $b$. So the endpoints of $b$ must be the same in the lift. Of the covers that you listed, only two have the property that the lifts of $b$ are loops: the disconnected cover and the cover with two lifts of $a$ running between two loops which are lifts of $b$. You can verify that if you attach two disks to this cover by $\bar a\bar b\bar a^{-1}$ for two disjoint sets of lifts $\bar a,\bar b$, then it really is a $2$-sheeted cover.


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