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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 27th 2014
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI've made comments at [[Fivebrane]], [[fivebrane 6-group]], [[Fivebrane structure]] and [[differential fivebrane structure]] regarding the fact $String(n)$ is not 6-connected for $n \leq 6$ (though trivially so for $n=2$). There will be a bunch of interesting invariants for manifolds of dimensions 3-6. This includes, for instance, on 6-dimensional spin manifolds a non-torsion class corresponding to the obstruction to lifting the tangent bundle to the 6-connected group covering $String(6)$ (here $\pi_5(String(6)) = \mathbb{Z}$). This is the only non-torsion example, but should be given by a $U(1)$-4-gerbe, I think, which will have a 6-form curvature. Since $H^6$ won't be vanishing for oriented manifolds, one has some checking to do. For instance, the frame bundle of $S^6$ lifted to a $String(6)$-bundle (I plan to write a paper on this 2-bundle) will not lift to a $Fivebrane(6)$-bundle, because it won't even lift to a $\widetilde{String(6)}$-bundle (i.e. the 6-connected cover of $String(6)$), since the transition function $S^5 \to BString(6)$ is the generator. Thus the 6-form curvature of this 4-gerbe should be the volume form on the 6-sphere. Another point that occurs to me is that there are two copies of $\mathbb{Z}$ to kill off in $String(8)$ to get $Fivebrane(8)$, so one gets a $U(1)\times U(1)$ higher gerbe. I suspect this larger $\pi_7$ is why there are so many more exotic spheres in dimension 15 than in neighbouring dimensions (16256 vs 2 in d=14 and 16); it's certainly the case for exotic spheres in dimension 7 that $\pi_3(SO(4)) = \mathbb{Z}\times \mathbb{Z}$ helps.

   I’ve made comments at Fivebrane, fivebrane 6-group, Fivebrane structure and differential fivebrane structure regarding the fact $String(n)$ is not 6-connected for $n \leq 6$ (though trivially so for $n=2$).

   There will be a bunch of interesting invariants for manifolds of dimensions 3-6. This includes, for instance, on 6-dimensional spin manifolds a non-torsion class corresponding to the obstruction to lifting the tangent bundle to the 6-connected group covering $String(6)$ (here $\pi_5(String(6)) = \mathbb{Z}$). This is the only non-torsion example, but should be given by a $U(1)$-4-gerbe, I think, which will have a 6-form curvature. Since $H^6$ won’t be vanishing for oriented manifolds, one has some checking to do. For instance, the frame bundle of $S^6$ lifted to a $String(6)$-bundle (I plan to write a paper on this 2-bundle) will not lift to a $Fivebrane(6)$-bundle, because it won’t even lift to a $\widetilde{String(6)}$-bundle (i.e. the 6-connected cover of $String(6)$), since the transition function $S^5 \to BString(6)$ is the generator. Thus the 6-form curvature of this 4-gerbe should be the volume form on the 6-sphere.

   Another point that occurs to me is that there are two copies of $\mathbb{Z}$ to kill off in $String(8)$ to get $Fivebrane(8)$, so one gets a $U(1)\times U(1)$ higher gerbe. I suspect this larger $\pi_7$ is why there are so many more exotic spheres in dimension 15 than in neighbouring dimensions (16256 vs 2 in d=14 and 16); it’s certainly the case for exotic spheres in dimension 7 that $\pi_3(SO(4)) = \mathbb{Z}\times \mathbb{Z}$ helps.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 27th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for adding that! Long overdue.

   Thanks for adding that! Long overdue.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 27th 2014
   • (edited Nov 27th 2014)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexRegarding my last comment regarding exotic 15-spheres, lemma 2 in <http://www.maths.ed.ac.uk/~aar/papers/shimada.pdf> is relevant. And I now believe the 992 exotic spheres in dimension 11 are due, at least in part, to $\pi_5(String(6)) = \mathbb{Z}$. This is pretty cool, if this is really higher bundles 'detecting' exotic spheres.

   Regarding my last comment regarding exotic 15-spheres, lemma 2 in http://www.maths.ed.ac.uk/~aar/papers/shimada.pdf is relevant. And I now believe the 992 exotic spheres in dimension 11 are due, at least in part, to $\pi_5(String(6)) = \mathbb{Z}$. This is pretty cool, if this is really higher bundles ’detecting’ exotic spheres.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 27th 2014
   • (edited Nov 27th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThis sounds interesting. Could you elaborate just a tad more on this comment? There is that article by Witten arguing about gravitational instantons via exotic spheres and it was somehow related to the ninebrane story. But I need to remind myself. Not now, though, now I need to rush off.

   This sounds interesting. Could you elaborate just a tad more on this comment? There is that article by Witten arguing about gravitational instantons via exotic spheres and it was somehow related to the ninebrane story. But I need to remind myself. Not now, though, now I need to rush off.

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 28th 2014
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexHmm, there's a recent question [on physicsoverflow](http://www.physicsoverflow.org/20309/exotic-spheres-in-string-m-theory) that asks whether these ideas are still up for grabs. If we consider $S^3$-fibre bundles over $S^4$ with structure group $SO(4)$, they are classified by $[S^3,SO(4)]=\mathbb{Z}^2$. Given $(n,m)$ describing such a map, if $n+m=1$ then the total space of the bundle is homeomorphic to a 7-sphere, and $\frac{p_1}{2} = n-m$. The differentiable structure is only standard for $(n-m)^2 - 1 = 0 (mod 7)$. I wish I knew what that mod 7 was meant to be. It comes in Milnor's proof because of the factor of 7 in Hirzebruch's signature theorem for 8-manifolds, but that's not immensely satisfying. I don't know how to describe the exotic spheres in dimension 11 yet.

   Hmm, there’s a recent question on physicsoverflow that asks whether these ideas are still up for grabs.

   If we consider $S^3$-fibre bundles over $S^4$ with structure group $SO(4)$, they are classified by $[S^3,SO(4)]=\mathbb{Z}^2$. Given $(n,m)$ describing such a map, if $n+m=1$ then the total space of the bundle is homeomorphic to a 7-sphere, and $\frac{p_1}{2} = n-m$. The differentiable structure is only standard for $(n-m)^2 - 1 = 0 (mod 7)$. I wish I knew what that mod 7 was meant to be. It comes in Milnor’s proof because of the factor of 7 in Hirzebruch’s signature theorem for 8-manifolds, but that’s not immensely satisfying.

   I don’t know how to describe the exotic spheres in dimension 11 yet.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeNov 28th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexAh, thanks! If you now add a pointer to the literature to this paragraph, then that's something to be copied into some $n$Lab entry.

   Ah, thanks! If you now add a pointer to the literature to this paragraph, then that’s something to be copied into some $n$Lab entry.