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?

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- Oct 12th 2009, 08:28 PMmoohe12Non normal Cover
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? - Oct 13th 2009, 06:33 PMaliceinwonderland
For an k-sheeted covering space (finite-degree covering map), $\displaystyle \tilde{M} \rightarrow M$, there is an Euler characteristic equation,

$\displaystyle \chi(\tilde{M}) = k \cdot \chi(M)$.

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

$\displaystyle \pi_1(K) = <a, b | abab^{-1}>$.

Because you are looking for a non-normal covering space, you need to find a non-normal subgroup H of $\displaystyle G=\pi_1(K)$. Recall that if a subgroup H of G is normal, then $\displaystyle xHx^{-1}=H$ for each x in G.

Now we try several cases:

1. H=<a, b^2>: This is not the case since $\displaystyle bab^{-1}=a^{-1} \in H$, where $\displaystyle a \in H$ and $\displaystyle b \in G$. 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, $\displaystyle \pi_1(H) = \mathbb{Z} \times \mathbb{Z}$ 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 $\displaystyle a^3$ and b such that $\displaystyle a^3b^2$ commutes ($\displaystyle a^3b^2 = b^2a^3$). You see that G is generated by a and b such that ab^2 commutes ( $\displaystyle ab^2 = abb=ba^{-1}b=bba=b^2a$).