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So I know the fact that the join of $S^1$ and $S^1$ is homeomorphic to the 3-sphere, but I'm having trouble "seeing" this. I'd prefer something that appeals to geometric intuition, but more formal abstract arguments are welcome as well! I want to invoke the unit quaternions here, but I can't think of a particularly simple way to do it.

If this question is too "soft" I sincerely apologize.

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    @AgustíRoig Methinks in general, S^m joined to S^n is homeomorphic to S^(m+n+1)...2012-09-20
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    The join in question is not $\vee$ but $*$: see [this](http://en.wikipedia.org/wiki/Join_%28topology%29) (One calls $\vee$ a wedge in this context, and dies a little every time one types `\wedge` to get a smash...)2012-09-20
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    @AgustíRoig Methinks you're wedging and not joining...2012-09-20
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    Ok, I mistook the "join" for the "wedge". Sorry.2012-09-20
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    Thanks for your explanations, Asingle and Mariano.2012-09-20

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