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Thread: Covering Spaces

  1. #1
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    Covering Spaces

    Hi it would be of great assistance if :

    Use Galois correspondence to find all the covering spaces of
    S^1 x S^1 x RP^2 (real projective plane is RP^2)

    Any help would be great. Thanks
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  2. #2
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    Quote Originally Posted by sorrow View Post
    Hi it would be of great assistance if :

    Use Galois correspondence to find all the covering spaces of
    S^1 x S^1 x RP^2 (real projective plane is RP^2)

    Any help would be great. Thanks
    Let $\displaystyle X = S^1 \times S^1 \times RP^2 $.

    Since X is semi-locally simply connected, there exists a Galois correspondence between subgroups of the fundamental group $\displaystyle \pi_1(X)$ and path-connected covering spaces of X.

    The fundamental group of X is $\displaystyle \pi_1(S^1 \times S^1 \times RP^2) =\pi_1(S^1) \times \pi_1(S^1) \times \pi_1(RP^2) = \mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}/2$.

    We shall find all the subgroups of $\displaystyle \pi_1(X)=\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}/2$ and connect each subgroup to its corresponding path-connected covering space of X.

    ("A subgroup of $\displaystyle \pi_1(X)$" $\displaystyle \longrightarrow$ "A corresponding path-connected covering space of X")

    1. {e} $\displaystyle \longrightarrow \mathbb{Re} \times \mathbb{Re} \times S^2 $. This is a universal cover of X.
    2. $\displaystyle \{e\} \times \{e\} \times \mathbb{Z}/2 \longrightarrow\mathbb{Re} \times \mathbb{Re} \times RP^2 $.
    3. $\displaystyle \{e\} \times \mathbb{Z} \times \mathbb{Z}/2 \longrightarrow\mathbb{Re} \times S^1 \times RP^2 $.
    4. $\displaystyle \mathbb{Z} \times \mathbb{Z} \times \{e\} \rightarrow S^1 \times S^1 \times S^2 $.
    5. $\displaystyle n\mathbb{Z} \times m\mathbb{Z} \times \{e\} \rightarrow S^1 \times S^1 \times S^2 $, where n and m are positive integers greater than 1.
    In this case, consider a covering map $\displaystyle p_n:S^1 \rightarrow S^1$ given by $\displaystyle p_n(1, \theta) = (1, n\theta)$, where $\displaystyle (r, \theta)$ is a polar coordinate in the plane $\displaystyle \mathbb{Re}^2$. The map $\displaystyle p_n$, where n is a positive integer, wraps the circle around itself n times. Thus $\displaystyle (S^1, p_n)$ is a covering space of $\displaystyle S^1$, corresponding a fundamental group $\displaystyle n\mathbb{Z}$.

    6. $\displaystyle n\mathbb{Z} \times \{e\} \times \mathbb{Z}/2 \rightarrow S^1 \times \mathbb{Re}^1 \times RP^2 $, where n is a positive integer greater than 1. It is similar to (5).

    7. $\displaystyle \mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}/2 \longrightarrow X$ itself.

    This problem becomes much more difficult if $\displaystyle X = S^1 \vee S^1 \vee RP^2$. It is because free product groups are involved in the fundamental group of X, and it is not easy to find all subgroups of a free product group and draw its Cayley graph.
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