Is there a good way to think about/understand the result that the double suspension of a homology 3-sphere is homeomorphic to a sphere, to get intuition for why this is true?  For instance, what sort of neighborhood must one take for one of the suspension vertices in the $S^0$ in order to see that a neighborhood of this point is homeomorphic to ${\mathbb R}^5$?  Thanks!! 

I found the following related MathOverflow question, which raises relevant points and is interesting, but doesn't seem to answer my question:

<a href="https://mathoverflow.net/questions/64029/if-a-manifold-suspends-to-a-sphere">"If a manifold suspends to a sphere..."</a>


My posting of this question today was partly inspired by thinking about the comments to the very recent MO question:

<a href="https://mathoverflow.net/questions/99295/is-a-finite-cw-complex-minus-a-point-still-homotopy-equivalent-to-a-finite-cw-com">"Is a finite CW complex minus a point still homotopy equivalent to a finite CW complex?"</a>

At some point, I asked the question I'm now posting to a topologist who is known for his incredible intuition, and he remarked that he believed it was better to think in terms of taking a join with $S^1$ rather than repeatedly taking a join with $S^0$, but he wasn't sure what else to say.

Thanks again for any help with this!