Let $X$ be a path-connected, locally path-connected and semilocally simply-connected space. Can we find a correspondence between degree $n$ covering spaces of $X$ and group homomorphism $\pi_1(X)\rightarrow S_n$? ($S_n$ is the permutation group)
Identifying the numbers of degree $n$ covering spaces of $X$
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algebraic-topology
covering-spaces
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2From the classification of covering spaces of such a space, we know they are in correspondence with the subgroups of the fundamental group. Can you relate the index of the subgroup with some useful parameter of the covering? – 2012-08-21
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2Don't the subgroups of index n correspond to connected coverings only? – 2012-08-21
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0Connected $n$ covering spaces are in correspondence with the orbits of all index $n$ subgroup acted by conjugation. I cannot find an easy way to identify these orbits. – 2012-08-21