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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
Sn≃SO(n+1)/SO(n)and spin groups
Sn≃Spin(n+1)/Spin(n)and pin groups
Sn≃Pin(n+1)/Pin(n).added some more words to the section Coset space structure
added this reference in the section about group actions on spheres:
added pointer to
which according to p.2 of
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
SO(n)/SO(n−1)≃Sn−1U(n)/U(n−1)≃S2n−1SU(n)/SU(n−1)≃S2n−1Sp(n)/Sp(n−1)≃S4n−1Sp(n)⋅SO(2)/Sp(n−1)⋅SO(2)≃S4n−1Sp(n)⋅Sp(1)/Sp(n−1)⋅Sp(1)≃S4n−1G2/SU(3)≃S6Spin(7)/G2≃S7Spin(9)/Spin(7)≃S15added full publication data for
Thanks for bringing this up.
Checking the entry, I see that the condition missing in the statement of the Montgomery-Samelson theorem (here) was compactness (it was correctly stated in the References-section, though!).
I have added that missing condition in and then started a stub for a subsection (now here) on celestial spheres acted on by Lorentz groups.
Just a stub, please feel invited to expand.
Added propositions about the topological complexity of spheres as well as products of spheres (including tori as a special case).
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