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

    and spin groups

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

    and pin groups

    S nPin(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:


    SO(n)/SO(n1) S n1 U(n)/U(n1) S 2n1 SU(n)/SU(n1) S 2n1 Sp(n)/Sp(n1) S 4n1 Sp(n)SO(2)/Sp(n1)SO(2) S 4n1 Sp(n)Sp(1)/Sp(n1)Sp(1) S 4n1 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.
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