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.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 29th 2014

    I added a brief description of how the exotic 7-spheres are constructed at exotic smooth structure.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2014

    Thanks! I made it part of an Examples-section

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2017
    • (edited Jun 26th 2017)

    I have re-arranged the material at exotic smooth structure. Previously there had been duplications and the ordering of the examples had been a bit weird. I have merged what used to be called “Properties” and “Examples” into a single new section “Existence and Examples” with a bunch of subsections to ease navigation.

    Otherwise I didn’t edit the content.

    One question: the article by Stallings here keeps being referred to in our entry as authored by Stallings and Zeeman. But when I check it out (pdf) it seems to be unambiguously the case that Stallings is the single author, while Zeeman was only the communicating editor. If there is some hidden reason why Zeeman needs to be cited as a co-author, then this ought to be made explicit in the entry, otherwise the reference to Zeeman ought to be removed.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2017

    We have the sentence

    Then Kervaire and Milnor (1963) proved that there are only finitely many exotic smooth structures on all spheres in dimension 5 or higher.

    Can we say this more precisely? I suppose the sentence means to say that for each n5n \geq 5 there is a finite set of smooth structures on S nS^n. What about the existence of exotic smooth structures? I.e. for which n5n \geq 5 is it known that this finite set has Cardinality larger than 1?

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 26th 2017

    Yes, that’s right. We even know the list of odd-dimensional spheres with a unique smooth structure; only in dimensions 1, 3, 5, 61 (that paper is to appear in the Annals). Similarly, there is only a few such in even dimensions, but there is still the open case of S 126S^{126}.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 26th 2017

    The author who put in Zeeman (Torsten Asselmayer-Maluga) seems not to visit regularly, so I removed Zeeman.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 26th 2017

    Perhaps the association of names comes from the Stallings–Zeeman theorem.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2017
    • (edited Jun 26th 2017)

    David R., thanks!! I have now added some more statements from Wang-Xu 16 to the entry, here .

    In view of such recents results it sounds strange that our entry goes on to claim in the next subsection (here) that a “complete classification” of exotic smooth structures in dimension n5n \geq 5 has been given by Kirby and Siebenmann in 1977. What is it really that they proved?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2017

    Todd, David C.,okay, thanks!

    I have also added a warning at the beginning of the references-section on applications to physics (here).

    I haven’t looked at “Exotic smoothness and astrophysics” yet, for instance, but it sounds dubious. I think most of these references were added by T. A-M. I know that once there were some more among these, which I had removed when somebody pointed out to me that they seemed dubious.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 26th 2017

    Re #5, isn’t the 126 issue about the (related) Kervaire invariant problem?

    Regarding exotic smooth spheres, Milnor writes:

    the differentiable Poincaré hypothesis is true in dimensions 1, 2, 3, 5, 6, and 12, but unknown in dimension 4. I had conjectured that it would be false in all higher dimensions. However, Mahowald has pointed out that there is at least one more exceptional case: The group S 61S_61 is also trivial.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2017

    David C., and 56. That’s corollary 1.15 of the article which David R. just pointed to (Wang-Xu 16). Whatever else is to be said on this point should be added in this section here.

    • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 26th 2017

    Urs #8 I think it’s a classification up to knowing some other groups we might not have calculated (there are ingredients like the image of the J-homomorphism, the Kervaire invariant and so on). In particular, now I check Wang-Xu, the final classification in even dimensions is not 100%, but we know everything up to d=62 at least.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2017

    Okay, thanks.

    the final classification in even dimensions is not 100%, but we know everything up to d=62 at least.

    Yes, that’s how I have put it into the entry. They conjecture that it’s complete for all d5d \geq 5.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2017

    Is there an example of an exotic smooth structure whose construction and proof of exotic-ness would be elementary for readers with a background (just) in basic point-set topology?

    • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 26th 2017

    Probably those on the 7-sphere. Here is an undergraduate project describing them, supervised by May.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2017

    Thanks, that’s pretty good.

    • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 28th 2017

    I wonder if there might be some targets for smooth HoTT theorem provers in this work around exotic spheres.