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I was wondering if anyone visiting could help me figure out how to prove the following exercises from Ch.11 of Fulton's Algebraic Topology: A First Course.

(1) Show that any two-sheeted covering has a unique structure of $G$-covering, where $G = \mathbb{Z}/\mathbb{2Z}$ (in this case the group action is even ($=$ free and properly discontinuous)).

I am unclear about what it would mean for a $G$-covering to have a (unique) structure. If someone could help me get clear on this, that might set me straight!

(2) If $p: Y \rightarrow X = Y/G$ (the quotient induced by an even group action of $G$ on $Y$) is a $G$-covering that is trivial as a covering, show that it is isomorphic to the trivial $G$-covering.

Per the text's definition of "isomorphism between $G$-coverings," I expect that the key to this problem is finding a suitable homeomorphism (which I am terrible at in general)--once that is established, the rest should be easy enough. I guess I am at a point in problem solving where I have these new definitions but don't see how to use them effectively. So, if anyone had any suggestions about this, or even a suitable homeomorphism, I would really appreciate this.

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up vote 3 down vote accepted

For the first question, the author is handing you a covering and asking you to put an action of the group $G = \mathbb{Z}/2$ on it. To give this covering the structure of a $G$-cover, I just need to tell you how the group $G$ acts on the fibers over each point. In this case, since the cover is two-sheeted, there's really only one thing I can possibly do: I'll let the nontrivial element swap the elements in the fiber. This is the sense in which the structure is unique: once we start looking for ways to get $G$ to act on the cover, we find that there's only one thing we can do. From this point you can go and check that this satisfies all the underlying topological conditions (evenness, etc.)

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