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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2019

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

    diff, v39, current

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

    and spin groups

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

    and pin groups

    S n≃Pin(n+1)/Pin(n). S^n \;\simeq\; Pin(n+1)/Pin(n) \,.

    diff, v40, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 26th 2019
    • (edited Mar 26th 2019)

    added some more words to the section Coset space structure

    diff, v42, current

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

    added this reference in the section about group actions on spheres:

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

    diff, v45, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2019

    added pointer to

    • 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 nn-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) ≃S n−1 U(n)/U(n−1) ≃S 2n−1 SU(n)/SU(n−1) ≃S 2n−1 Sp(n)/Sp(n−1) ≃S 4n−1 Sp(n)⋅SO(2)/Sp(n−1)⋅SO(2) ≃S 4n−1 Sp(n)⋅Sp(1)/Sp(n−1)⋅Sp(1) ≃S 4n−1 G 2/SU(3) ≃S 6 Spin(7)/G 2 ≃S 7 Spin(9)/Spin(7) ≃S 15 \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}

    diff, v46, current

    • 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

    diff, v58, current

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

    added full publication data for

    • S. Gutt, Killing spinors on spheres and projective spaces, p. 238-248 in: A. Trautman, G. Furlan (eds.) Spinors in Geometry and Physics – Trieste 11-13 September 1986, World Scientific 1988 (doi:10.1142/9789814541510, GBooks, p. 243)

    diff, v59, current

  9. Guest #6 is right about the nonmetrizability of the CW-complex S ∞S^\infty, so I added some CW-facts which show this.

    Anonymous

    diff, v62, current

    • CommentRowNumber10.
    • CommentAuthorGuest
    • CommentTimeJan 19th 2023
    Implicit in this page seems to be the assumption that there is a Riemannian structure on the sphere. If we view the sphere under conformal transformations for example that gives other Lie groups acting on the sphere effectively and transitively. Specifically S^n can be realised as the quotient of (the identity component of) SO(n+1,1) by a parabolic subgroup.
    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2023
    • (edited Jan 19th 2023)

    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.

    diff, v65, current

  12. Fix typo

    Mark John Hopkins

    diff, v67, current

    • CommentRowNumber13.
    • CommentAuthorSamuel Adrian Antz
    • CommentTimeFeb 14th 2024
    • (edited Feb 14th 2024)

    Added propositions about the topological complexity of spheres as well as products of spheres (including tori as a special case).

    diff, v71, current

1 to 13 of 13

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