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Re: a covering map
Posted:
Mar 27, 2006 9:08 AM
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>>No it's not. That why the branches, infinitely many.
>>A covering that's not a universal covering is
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>> ___|___
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>> ______|______|_______
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>> | ~~~~~~~~
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>>With infinite iteration for the crosses as previously indicated.
>Where the square at the lower right is supposed to be a closed
>loop around one of the circles. I guess that's a covering map,
>I think probably.
Yes, it is.
>Now say H is a subgroup of G. Then the quotient space
>S/H is also a covering space of X, with covering map
> xH |-> pi(x).
>I seem to recall that every (connected>) covering space
>of X is of the form S/H for some such H, and this is a
>reason for calling (S, pi) the "universal" cover.
>But that may well be wrong. For example I don't see why
>the map you specify above is not a covering map, and I
>don't see how it arises as S/H.
Let the basepoint be at corner of the cirlce that also links
to the crosses.
Let A denote the deck transformation on the universal cover
that moves everything 'right' by one vertex. Then H will be the
subgroup generated by A. The circle is the image of the line
segment between the basepoint and the next vertex right. The
crosses are the image of the upper and lower crosses in the
universal cover.
--Dan Grubb
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