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 comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite 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 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).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2021

    added the statement that

    The stable tangent bundle of a unit sphere bundle S(𝒱)S(\mathcal{V}) in a real vector bundle 𝒱pM\mathcal{V} \overset{p}{\longrightarrow} M (Example \ref{UnitSphereBundles}) over a smooth manifold MM is isomorphic to the pullback of the direct sum of the stable tangent bundle of the base manifold with that vector bundle:

    T stabS(𝒱)S(p) *(T stabM M𝒱). T^{stab} S(\mathcal{V}) \; \simeq \; S(p)^\ast \big( T^{stab} M \oplus_M \mathcal{V} \big) \,.

    Still need to add a more canonical reference and/or a proof.

    diff, v3, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2021

    added pointer to p. 403 in

    where this statement appears somewhat between the lines.

    diff, v4, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2021
    • (edited Mar 24th 2021)

    I have written out (here) a proof of this Milnor-trivial statement

    diff, v5, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2021
    • (edited Mar 25th 2021)

    I have spelled out (here) a purely homotopy-type theoretic proof that the once-stabilized vertical tangent bundle to a sphere bundle associated to a vector bundle is the pullback of that vector bundle.

    (This is, somewhat implicitly, from Sec. 3 of our Twisted Cohomotopy implies M5-brane anomaly cancellation. Making it more explicit now in v2.)

    Incidentally, the tikzd diagrams don’t all come out scaled quite as intended: it seems that scaling just the columns with, say, [colum sep=tiny], scales also the rows, here on the nnLab.

    diff, v8, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2021

    added a concluding remark, to highlight:

    Prop. \ref{StableTangentBundleOfUnitSphereBundle} implies that every stable characteristic class of the tangent bundle of an orthogonal sphere-fiber bundle – i.e all polynomials in its Pontryagin classes – are basic, i.e. pulled back from the base space.

    diff, v10, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 16th 2022

    added pointer to today’s

    diff, v13, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 15th 2023

    added pointer to:

    diff, v14, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2023

    added pointer to:

    diff, v15, current