nLab exotic 7-sphere

Idea

A topological 7-sphere equipped with an exotic smooth structure is called an exotic 7-sphere.

Milnor’s construction

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 modulo-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).

Examples

• Gromoll-Meyer sphere

Properties

As near-horizon geometries of black M2-branes

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, 3.8))

References

General

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

• Egbert Brieskorn, Beispiele zur Differentialtopologie von Singularitäten, Inventiones mathematicae 2 (1966) 1–14 $[$doi:10.1007/BF01403388$]$

• 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)

• Niles Johnson, Visualizing 7-manifolds, 2012 (nilesjohnson.net/seven-manifolds.html)

• Diarmuid Crowley, Christine Escher, A classification of $S^3$-bundles over $S^4$, Differential Geometry and its Applications Volume 18, Issue 3, May 2003, Pages 363-380 (doi:10.1016/S0926-2245(03)00012-3))

See also

• Wikipedia, Exotic sphere

In M-theory

• David Morrison, M. Ronen Plesser, section 3.2 of Non-Spherical Horizons, I, Adv. Theor. Math. Phys. 3:1-81, 1999 (arXiv:hep-th/9810201)