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I just started to learn for an exam but I am stuck in this exercise: Let Y be a topological space with $\pi_1(X)=\mathbb{Z}/_{19}$ . Is there a covering space (order 4) ? How do you construct a covering space in this case? Thank you very much!

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You're given information about the fundamental group of $Y$, and want to get information about the covering spaces of $Y$. You should review what you've learned about the relationship between fundamental groups and covering spaces. –  Chris Eagle Nov 17 '11 at 22:43
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Hint: A connected covering space of $Y$ of order $4$ would correspond to a subgroup of $\pi_1(Y)$ of index $4$.

If you don't require connectedness then it is easy to find a covering space of any order.

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@kobloau: Do you know about Lagrange's theorem in finite group theory? –  Zhen Lin Nov 20 '11 at 0:11
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