Added

]]>There exists an infinite family of mutually non-diffeomorphic irreducible smooth structures on the topological 4-manifold $S^2 \times S^2$ (Akhmedov-Park 10).

Added something on the exotic 8-sphere (h/t Jack Morava in reaction to my mention of ’Twisted Cohomotopy implies M-theory anomaly cancellation on 8-manifolds’)

]]>The abelian group of non-diffeomorphic structures with connected sum on the 8-sphere is the cyclic group $Z/2$. The unique exotic 8-sphere corresponds to the nontrivial element of the cokernel of the J-homomorphism and is the first instance of an exotic sphere that does not bound a parallelizable manifold (Amabel 17, Sec. 11). It admits a metric of positive Ricci curvature.

added DOI and pdf-link for:

- John Milnor,
*On manifolds homeomorphic to the 7-sphere*, Annals of Mathematics (Annals of Mathematics) 64 (2): 399–405 (1956) (pdf, doi:10.1142/9789812836878_0001)

I know, the page should give only the idea

No, this is not a demand we have on the page! It’s just that so far nobody had the energy to do more.

I add some technical things in a general style.

Good, thanks. If you feel like expanding this entry to a more comprehensive technical discussion, that would be great. Maybe have a look at our template page for inspiration.

]]>The reference list for the applications to physics is nearly complete. I have no idea what else. I remember on a proceeding paper by Penrose quite after the appearance of Donaldson reults (in 1983). There, he speculates about exotic smoothness and its relation to physics vis the anti-self-dual solutions. I will look for that paper.

There are two constructions for exotic compact 4-manifolds: the knot surgery and the Akbulut cork which should be discussed....

Or is the article to long?

Torsten ]]>

A little bit more, but I’ll just leave links to Wikipedia pages for references (of which there are a number) at the moment

]]>I suggest

I have now created a “category: people”-entry Jerzy Krol and for the moment moved his list of publications to there.

]]>Oh, and I created a stub for topological manifold directing to manifold (where there is detailed discussion).

]]>Thanks. I edited it a bit, check if you like it.

Maybe you find a second to add the references that you allude to to the References-section. That would serve to counterbalance the huge list of potential physical applications with a little bit of genuine substance :-)

]]>Something mathematical at exotic smooth structure, namely an idea section and reference to Milnor’s 7-sphere paper.

]]>many more references added

There is this long list of references by Jerzy Krol (a whole subsection). Did anyone look at this?

I think the majority of the references by him have nothing to do with exotic smooth structure – and don’t claim to have. It’s just the two joint papers with Torster Asselmeyer-Maluga that do.

So I think that section with references of Krol should not be in this entry.

In this long series of articles currently listed at exotic smooth structure Krol speculates aout possible applications of topos theory and other abstract category theory to physics. For instance he asks

whether at small scales spacetime might have to be modeled by a general topos instead of a topological space,

whether model theory might explain the AdS/CFT correspondence,

whether non-standard integers in the Basel topos of synthetic differential geometry might explain how to handle divergencies in quantum field theory.

Things like that. I think these are in principle interesting questions (though I don’t understand what he means in the AdS/CFT case) but that they remain very speculative.

I would suggest we move this list of publications by Krol to a different entry, say to n-categorical physics and add comments indicating that this is speculative.

]]>Yes, lots of references added by Torsten, and thanks for them!

Tip: Instead of creating Torsten Asselmeyer, add `[[!redirects Torsten Asselmeyer]]`

to the bottom of Torsten Asselmeyer-Maluga. (I have fixed this now.)

Torsten Asselmeyer-Maluga added a reference to an article of his

]]>recorded two references to physics application at exotic smooth structure

(no time for anything else)

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