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.