Hi I am try to solve Hatcher problem 20 section 1.3:
Find a non-normal Covering spaces of the Klein bottle by a Klein bottle and by a torus.
Anyone have a hint to start with?
For an k-sheeted covering space (finite-degree covering map), , there is an Euler characteristic equation,
.
The Euler characteristic of a torus and a klein bottle are both zero. Thus we see that a torus or klein bottle itself can be a covering space for a klein bottle.
The fundamental group of a klein bottle can be presented as
.
Because you are looking for a non-normal covering space, you need to find a non-normal subgroup H of . Recall that if a subgroup H of G is normal, then for each x in G.
Now we try several cases:
1. H=<a, b^2>: This is not the case since , where and . H is normal in G.
2. H=<a^2, b> : This is the case since we cannot find x in G such that xHx^-1 = H.
3. H=<a^3, b> : This is the case since we cannot find x in G such that xHx^-1 = H.
....
I recommend you try several more cases.
For 1, that is the fundamental group of a torus, but it is not a normal subgroup of G.
For 3, H corresponds to the klein bottle since H is generated by and b such that commutes ( ). You see that G is generated by a and b such that ab^2 commutes ( ).