Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: sphere

Bottom of Page

1 to 13 of 13

  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

  1.