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An Algebraic Topology question: covering spaces Watch

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    Hello there,

    I'm stuck on a particular algebraic topology question. Here goes.

    I am given a space X:=S_1\vee S_1, the wedge of two circles. This has fundamental group \pi_1(K,b)=\langle x,y\rangle. The plan is to find a based covering map p: (\tilde{X},\tilde{b}) \rightarrow (X,b) such that p_{\star}\pi_1(\tilde{x},\tilde{  b}) is the subgroup generated by x.

    I believe this would require me to find a covering space whose fundamental group was isomorphic to the group generated by x, i.e. to \mathbb{Z}, but I'm having no luck. Any ideas?

    Thanks in advance,

    M
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    Couldn't you just cover the x-circle with a circle and the y-circle with a line - so the covering space is the wedge of a circle and a line?
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    RichE,

    Thanks for the reply. This would appear not to fit with the definition of a covering space, for example at http://www.maths.ox.ac.uk/system/fil...tg050908_0.pdf page 56. The problem is at the single vertex: its preimage under p needs to have an open neighbourhood in \tilde{X} homeomorphic to four copies of [0,1) joined at 0.

    Also, I have a feeling we need a covering space with infinitely many vertices, since \mathbb{Z} has (I believe) infinite index in the free group on two generators. I have no inspiration!
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    (Original post by mpd1989)
    RichE,

    Thanks for the reply. This would appear not to fit with the definition of a covering space, for example at http://www.maths.ox.ac.uk/system/fil...tg050908_0.pdf page 56. The problem is at the single vertex: its preimage under p needs to have an open neighbourhood in \tilde{X} homeomorphic to four copies of [0,1) joined at 0.

    Also, I have a feeling we need a covering space with infinitely many vertices, since \mathbb{Z} has (I believe) infinite index in the free group on two generators. I have no inspiration!
    Sorry, didn't notice that it would fail to be a covering map over ths vertex. How about instead you had the real line with a circle attached at each integer. The circles map onto the x-circle and the real line maps onto the y-circle.
 
 
 
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