As many of you know, using the stereographic projection one can construct a homeomorphism between the the complex plane ℂ(adsbygoogle = window.adsbygoogle || []).push({}); ^{1}and the unit sphere S^{2}∈ℝ^{3}. But the stereographic projection can be extended to

the n-sphere/n-dimensional Euclidean space ∀n≥1. Now what I am talking about is the the mappingI: H→ HwhereHis the space of all Quaternions and I(q) = (1/q) ∀q ∈H. So this complex manifold is the one-point compactification ofHwhich I will refer to as ◊.

That is, ◊ :H ∪ {∞}.I: 1/(0+0i+0j+0k) ↔ {∞}. So is there an official name for ◊ and has it already been shown that it is topologically equivalent to S^{4}? I assume so but if need be I will give a proof attempt in a followup post.

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# The hyperriemann sphere

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