Determine a normal covering space

Question

Is there a way to determine if a covering space is normal without using the two theorems of Hatcher's book in pages 71 and 72?

Answer 1

The only ways I know how to do this are to either enumerate the Deck transformations (which is of course not an algorithmic process, but maybe it's easy from the way you defined the covering map) and verify that they act transitively on the fibers, or use the result of Hatcher's you mention. I haven't seen anybody do anything else.

One thing that's worth noting is the case of double covers. These are automatically normal for both of the reasons above: if you want to think about subgroups, index 2 subgroups are automatically normal; if you want to think about deck transformations, the map that swaps each element of the fiber is continuous (look locally), bijective, and a local homeomorphism.

(If you follow through the proof of Hatcher's theorems, where you see how he calculates the deck transformation group, it will become obvious that these are the same fact. But I like the ability to look at it in two different lenses.)