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Math Help - Non normal Cover

  1. #1
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    Non 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?
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  2. #2
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    Quote Originally Posted by moohe12 View Post
    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), \tilde{M} \rightarrow M, there is an Euler characteristic equation,
    \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
    \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 G=\pi_1(K). Recall that if a subgroup H of G is normal, then xHx^{-1}=H for each x in G.

    Now we try several cases:
    1. H=<a, b^2>: This is not the case since bab^{-1}=a^{-1} \in H, where a \in H and 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, \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 a^3 and b such that a^3b^2 commutes ( a^3b^2 = b^2a^3). You see that G is generated by a and b such that ab^2 commutes ( ab^2 = abb=ba^{-1}b=bba=b^2a).
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