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:

"If a manifold suspends to a sphere..."

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

"Is a finite CW complex minus a point still homotopy equivalent to a finite CW complex?"

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!