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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 6th 2019

subdivided the Properties-section into subsections; added subsection for branched coverings of $n$-spheres

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMar 16th 2019
• (edited Mar 16th 2019)

have expanded the statement

The n-spheres are coset spaces of orthogonal groups

S^n \;\simeq\; O(n+1)/O(n),.\$

to :

Similarly for the corresponding special orthogonal groups

$S^n \;\simeq\; SO(n+1)/SO(n)$

and spin groups

$S^n \;\simeq\; Spin(n+1)/Spin(n)$

and pin groups

$S^n \;\simeq\; Pin(n+1)/Pin(n) \,.$
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMar 26th 2019
• (edited Mar 26th 2019)

added some more words to the section Coset space structure

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeApr 29th 2019
• (edited Apr 29th 2019)

• Alfred Gray, Paul S. Green, Sphere transitive structures and the triality automorphism, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 (euclid:1102976640)
• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeApr 29th 2019

• Deane Montgomery, Hans Samelson, Transformation Groups of Spheres, Annals of Mathematics Second Series, Vol. 44, No. 3 (Jul., 1943), pp. 454-470 (jstor:1968975)

which according to p.2 of

• Alfred Gray, Paul S. Green, Sphere transitive structures and the triality automorphism, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 (euclid:1102976640)

is the actual origin of the classification of the coset space realizations of $n$-spheres, which I have added now as a Prop:

The connected Lie groups with effective transitive actions on n-spheres are precisely (up to isomorphism) the following:

with

\begin{aligned} SO(n)/SO(n-1) & \simeq S^{n-1} \\ U(n)/U(n-1) & \simeq S^{2n-1} \\ SU(n)/SU(n-1) & \simeq S^{2n-1} \\ Sp(n)/Sp(n-1) & \simeq S^{4n-1} \\ Sp(n)\cdot SO(2)/Sp(n-1)\cdot SO(2) & \simeq S^{4n-1} \\ Sp(n)\cdot Sp(1)/Sp(n-1)\cdot Sp(1) & \simeq S^{4n-1} \\ G_2/SU(3) & \simeq S^6 \\ Spin(7)/G_2 & \simeq S^7 \\ Spin(9)/Spin(7) & \simeq S^{15} \end{aligned}
• CommentRowNumber6.
• CommentAuthorGuest
• CommentTimeNov 14th 2019
The directed colimit of finite dimensional spheres (endowed with the colimit space topology) is not the unit sphere in some normed space. The first one is a non-locally-finite CW-complex and hence not metrizable, while the second one is a subset of a normed space and hence a metrizable space.
• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMar 27th 2021

added mentioning of the canonical spin-structure on the n-sphere, induced from the coset-space realization

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeMar 28th 2021
• (edited Mar 28th 2021)