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$\begingroup$

I'm a little stuck at the moment where to go next with this. I know that there is a fact that there is a curve in $SO(3)$, beginning and ending at the identity which cannot be deformed to the constant curve, but when traversed twice can be deformed.

I want to be able to explain this clearly. Can someone explain this to me or help me out a little please?

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    Any addition infomation about this will be helpfull to.2012-05-07
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    Well, this is just a consequence of the fact that the fundamental group of SO(3) is the cyclic group of order 2. Equivalently, the universal cover of SO(3) is the group SU(2), and the kernel is $\{\pm 1\}$.2012-05-07
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    In fact, it is explained on this wiki page: http://en.wikipedia.org/wiki/Rotation_group_SO%283%29#Topology2012-05-07
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    This has nothing to do with Lie algebra...2012-05-07
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    thank you M Turgeon and Olivier2012-05-07
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    You may be interested in Dirac's *[Belt trick](http://virtualmathmuseum.org/Surface/dirac-belt/DiracBelt.pdf)*. See also [this question](http://math.stackexchange.com/questions/10916/)2012-05-07
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    @MTurgeon, it looks to me like he's asking for a proof of that fact ... .2012-05-07
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    @Neal ... thus the link2012-05-07
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    @MTurgeon Touche.2012-05-07

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