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I have calculated the fundamental group of sphere with a diameter using Van-Kampen theorem, which is $Z$. So corresponding to subgroup $2Z$ there exist a two sheeted connected covering space. So existence is clear. But I am not getting idea how to find it explicitly. Any help in this direction shall be appreciated.

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You can see that your space is homotopically equivalent to $S^1\vee S^2$ and the double cover space of $S^1\vee S^2$ is $S^2\vee S^1\vee S^2$.

This is not the double cover of your space but it is homotopically equivalent to that but it is easier to visualize. You can analogously construct the cover space of the sphere with diameter: you can see it as two copies of $S^2$ with say the corresponding north poles and south poles connected by an edge.

pic

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  • $\begingroup$ why this is not double cover of my space?I think this is a double cover of my space.... $\endgroup$ – Ripan Saha Mar 7 '15 at 7:33
  • $\begingroup$ I mean $S^2\vee S^1 \vee S^2$ is not the double cover of your space, but the one in the picture is. However these two spaces are homotopically equivalent (but not homeomorphic). $\endgroup$ – Dario Mar 7 '15 at 8:53
  • $\begingroup$ okk...I see..could you tell me how did you construct that space...Is it hit and trial or there is some procedure to construct covering space for some "good" topological space? $\endgroup$ – Ripan Saha Mar 8 '15 at 8:59
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    $\begingroup$ First I switched a homotopically equivalent space with a simple structure (wedge), i.e. $S^1\vee S^2$, then I constructed the cover space of that, and in the end I modified it in order to make it a cover space for the original space. It is pretty simple to construct the double cover of $S^1\vee S^2$: since $S^2$ is simply connected you just have to "stick" two copies of that to the double cover of $S^1$. $\endgroup$ – Dario Mar 8 '15 at 10:37
  • $\begingroup$ @Dario...thank you... $\endgroup$ – Ripan Saha Mar 8 '15 at 11:08

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