synthetic differential geometry
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$(ʃ \dashv \flat \dashv \sharp )$
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$ʃ_{dR} \dashv \flat_{dR}$
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$(\Re \dashv \Im \dashv \&)$
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$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
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A topological 7-sphere equipped with an exotic smooth structure is called an exotic 7-sphere.
Milnor (1956) gave the first examples of exotic smooth structures on the 7-sphere, finding at least seven.
The exotic 7-spheres constructed in Milnor 1956 are all examples of fibre bundles over the 4-sphere $S^4$ with fibre the 3-sphere $S^3$, with structure group the special orthogonal group SO(4) (see also at 8-manifold the section With exotic boundary 7-spheres):
By the classification of bundles on spheres via the clutching construction, these correspond to homotopy classes of maps $S^3 \to SO(4)$, i.e. elements of $\pi_3(SO(4))$. From the table at orthogonal group – Homotopy groups, this latter group is $\mathbb{Z}\oplus\mathbb{Z}$. Thus any such bundle can be described, up to isomorphism, by a pair of integers $(n,m)$. When $n+m=1$, then one can show there is a Morse function with exactly two critical points on the total space of the bundle, and hence this 7-manifold is homeomorphic to a sphere.
The fractional first Pontryagin class $\frac{p_1}{2} \in H^4(S^4) \simeq \mathbb{Z}$ of the bundle is given by $n-m$. Milnor constructs, using cobordism theory and Hirzebruch's signature theorem for 8-manifolds, a mod-7 diffeomorphism invariant of the manifold, so that it is the standard 7-sphere precisely when $\frac{p_1}{2}^2 -1 = 0 (mod 7)$.
By using the connected sum operation, the set of smooth, non-diffeomorphic structures on the $n$-sphere has the structure of an abelian group. For the 7-sphere, it is the cyclic group $\mathbb{Z}/{28}$ and Brieskorn (1966) found the generator $\Sigma$ so that $\underbrace{\Sigma\#\cdots\#\Sigma}_28$ is the standard sphere.
Review includes (Kreck 10, chapter 19, McEnroe 15, Joachim-Wraith).
From the point of view of M-theory on 8-manifolds, these 8-manifolds $X$ with (exotic) 7-sphere boundaries in Milnor’s construction correspond to near horizon limits of black M2 brane spacetimes $\mathbb{R}^{2,1} \times X$, where the M2-branes themselves would be sitting at the center of the 7-spheres (if that were included in the spacetime, see also Dirac charge quantization).
(Morrison-Plesser 99, section 3.2, FSS 19, 4.6))
Matthias Kreck, chapter 19 of Exotic 7-spheres of Differential Algebraic Topology – From Stratifolds to Exotic Spheres, AMS 2010
Rachel McEnroe, Milnor’ construction of exotic 7-spheres, 2015 (pdf)
Michael Joachim, D. J. Wraith, Exotic spheres and curvature (pdf)
See also
Last revised on March 2, 2020 at 10:41:06. See the history of this page for a list of all contributions to it.