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I am having great difficulty with the following qualifying exam problem and would appreciate some help. Thank you so much in advance.

Give an example of a (path-connected) covering space which is not a regular covering space.

kyle
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1 Answers1

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For sufficiently nice topological spaces, regular covering spaces correspond to normal subgroups of the fundamental group. So find any space with a fundamental group containing subgroups that aren't normal.

For a specific example, consider $X=S^1\vee S^1$, whose fundamental group is $$G=\mathbb{Z}*\mathbb{Z}=\langle a\rangle*\langle b\rangle$$ The subgroup $\langle a\rangle$ is not normal in $G$, and therefore corresponds to a covering space which is not regular.

Check Hatcher (pg. 58) for pictures of other nonregular covering spaces, including the one I've just described. On pg. 58, (3), (4), (9), (10), and (12)-(14) are all nonregular.

Jared
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  • So in this example would any cover of index 2 be an irregular covering? – kyle Apr 25 '13 at 22:03
  • Careful. $\langle a \rangle$ does not have index $2$. Any subgroup of index $2$ is in fact normal, and thus corresponds to a regular covering. In the link I gave, (1) and (2) are examples of coverings of index $2$, and both are regular. – Jared Apr 25 '13 at 22:11