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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 29th 2014
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI added a brief description of how the exotic 7-spheres are constructed at [[exotic smooth structure]].

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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 30th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks! I made it part of an Examples-section

   Thanks! I made it part of an Examples-section

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 26th 2017
   • (edited Jun 26th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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](https://ncatlab.org/nlab/show/exotic+smooth+structure#StallZee) keeps being referred to in our entry as authored by Stallings and Zeeman. But when I check it out ([pdf](http://www.maths.ed.ac.uk/~aar/papers/stallings2.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.

   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.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 26th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexWe 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 $n \geq 5$ there is a finite set of smooth structures on $S^n$. What about the existence of exotic smooth structures? I.e. for which $n \geq 5$ is it known that this finite set has Cardinality larger than 1?

   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 $n \geq 5$ there is a finite set of smooth structures on $S^n$. What about the existence of exotic smooth structures? I.e. for which $n \geq 5$ is it known that this finite set has Cardinality larger than 1?

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 26th 2017
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexYes, that's right. We even know the list of odd-dimensional spheres with a unique smooth structure; [only in dimensions 1, 3, 5, 61](https://arxiv.org/abs/1601.02184) (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 $S^{126}$.

   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 $S^{126}$.

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 26th 2017
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThe author who put in Zeeman (Torsten Asselmayer-Maluga) seems not to visit regularly, so I removed Zeeman.

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

7. • CommentRowNumber7.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 26th 2017
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexPerhaps the association of names comes from the [Stallings–Zeeman theorem](https://en.wikipedia.org/wiki/Stallings%E2%80%93Zeeman_theorem).

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

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJun 26th 2017
   • (edited Jun 26th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexDavid R., thanks!! I have now added some more statements from Wang-Xu 16 to the entry, [here](https://ncatlab.org/nlab/show/exotic+smooth+structure#ExoticNSpheresForNGreaterThanFour) . In view of such recents results it sounds strange that our entry goes on to claim in the next subsection ([here](https://ncatlab.org/nlab/show/exotic+smooth+structure#exotic_smooth_structures_in_dimension_)) that a "complete classification" of exotic smooth structures in dimension $n \geq 5$ has been given by Kirby and Siebenmann in 1977. What is it really that they proved?

   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 $n \geq 5$ has been given by Kirby and Siebenmann in 1977. What is it really that they proved?

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJun 26th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexTodd, David C.,okay, thanks! I have also added a warning at the beginning of the references-section on applications to physics ([here](https://ncatlab.org/nlab/show/exotic+smooth+structure#for_applications_to_physics)). 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.

   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.

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 26th 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexRe #5, isn't the 126 issue about the (related) Kervaire invariant problem? Regarding exotic smooth spheres, Milnor [writes](http://www.ams.org/notices/201106/rtx110600804p.pdf): >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 $S_61$ is also trivial_.

    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 $S_61$ is also trivial.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2017
    • PermaLink
    Author: Urs
    Format: MarkdownItexDavid C., and 56. That's corollary 1.15 of the article which David R. just pointed to ([Wang-Xu 16](https://ncatlab.org/nlab/show/exotic+smooth+structure#WangXu16)). Whatever else is to be said on this point should be added in this section [here](https://ncatlab.org/nlab/show/exotic+smooth+structure#ExoticNSpheresForNGreaterThanFour).

    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.

12. • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 26th 2017
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexUrs #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.

    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.

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2017
    • PermaLink
    Author: Urs
    Format: MarkdownItexOkay, 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 $d \geq 5$.

    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 $d \geq 5$.

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2017
    • PermaLink
    Author: Urs
    Format: MarkdownItexIs 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?

    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?

15. • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 26th 2017
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexProbably those on the 7-sphere. [Here](http://math.uchicago.edu/~may/REU2015/REUPapers/McEnroe.pdf) is an undergraduate project describing them, supervised by May.

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

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2017
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks, that's pretty good.

    Thanks, that’s pretty good.

17. • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 28th 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexI wonder if there might be some targets for smooth HoTT theorem provers in this work around exotic spheres.

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