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I'm currently taking a course on algebraic topology and while doing exercises, I realised that I wanted to use the following:

If $X$ is a compact connected $2$-manifold and $\varpi:Y \rightarrow X $ a connected $2$-sheeted covering space, then $\chi (Y)=2 \cdot \chi(X)$.

Since $X$ admits a triangulation, I tried to reason on the triangulation that the number of faces, edges and vertices must double since $\varpi$ is 2-sheeted. But since I want it to be connected, I have to do some identifications and I don't know how to conclude.

I suppose that this is a particular case and that it's true with finite-sheeted coverings and spaces that don't have to be manifolds (related: Euler characteristic of a covering space )

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Javier
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    http://math.stackexchange.com/questions/1568094/euler-characteristic-of-covering-space-of-cw-complex/1568112#1568112 – Balarka Sen Dec 13 '15 at 20:41
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    The idea is to just use the lifting property of covering spaces to lift the inclusion map of each simplex (in two different ways), rather than trying to explicitly figure out how to glue them together in $Y$ by hand. – Eric Wofsey Dec 13 '15 at 20:57
  • Thank you for the answers. I searched the answer, but somehow overlooked that question. Also, I've just learnt what a CW structure is and it seems very useful. – Javier Dec 14 '15 at 18:37
  • @EricWofsey What is the lifting property of covering spaces? Is it the homotopy lifting property? Isn't that only for homotopies? – user5826 Oct 17 '19 at 00:53
  • @AlJebr: No, there is a much more general lifting property. See the paragraph starting "More generally, ..." at https://en.wikipedia.org/wiki/Covering_space#Lifting_properties, for instance. – Eric Wofsey Oct 17 '19 at 01:32
  • @EricWofsey I see now. But what if when we lift a simplex, it intersects the lifting of another simplex? Isn't this a problem? Say, $x$ is in some simplex, and $y$ is in some other simplex, wouldn't it be a problem if the lifting of the simplex containing $x$ intersects the lifting of the simplex containing $y$? – user5826 Oct 17 '19 at 22:01
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    @AlJebr: If two lifted simplices intersect, then the original simplices must also have intersected, since you can just compose with the covering map. So for instance, two lifted triangles can only intersect along a vertex or edge, and only if the original triangles did. – Eric Wofsey Oct 17 '19 at 22:06

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