Do there exist covering maps $p:X\rightarrow Y$ and $q:Y\rightarrow Z$ such that $X$ is path connected and the composition $q\circ p$ fails to be a covering map?
Composition of coverings of path connected spaces
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algebraic-topology
covering-spaces
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0The paper : J. Brazas, "semicoverings: a generalisation of covering space theory", Homology, Homotopy and Applications, 14 (2012) 33-63, shows that semicoverings satisfy the "$2$ out of $3$ property". So $q \circ p$ will be a semicovering! – 2012-06-27
1 Answers
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I think a cover of a cover of the Hawaiian earring gives an example where the composition fails to be a covering space, and the space $X$ is path conencted
See Exericse 6 on page 79 of Hatcher's book
Edit: The composition will be a covering map if the fiber $q^{-1}(z)$ is finite (proof) or, equivalently, the space $Z$ is semi-locally simply connected (In particular the Hawaiian earring is not semi-locally simply connected).
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0@Jacob - yes that is exactly what I meant! O$f$ course covering space fibers are always discrete... – 2011-07-29