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# An Algebraic Topology question: covering spaces Watch

1. Hello there,

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

I am given a space , the wedge of two circles. This has fundamental group . The plan is to find a based covering map such that 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 , but I'm having no luck. Any ideas?

M
2. 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?
3. 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 needs to have an open neighbourhood in homeomorphic to four copies of joined at 0.

Also, I have a feeling we need a covering space with infinitely many vertices, since has (I believe) infinite index in the free group on two generators. I have no inspiration!
4. (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 needs to have an open neighbourhood in homeomorphic to four copies of joined at 0.

Also, I have a feeling we need a covering space with infinitely many vertices, since 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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Updated: December 28, 2010
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