exotic 7-sphere in nLab

Contents

Contents

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 4S^4 with fibre the 3-sphere S 3S^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→SO(4)S^3 \to SO(4), i.e. elements of π 3(SO(4))\pi_3(SO(4)). From the table at orthogonal group – Homotopy groups, this latter group is ℤ⊕ℤ\mathbb{Z}\oplus\mathbb{Z}. Thus any such bundle can be described, up to isomorphism, by a pair of integers (n,m)(n,m). When n+m=1n+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 p 12∈H 4(S 4)≃ℤ\frac{p_1}{2} \in H^4(S^4) \simeq \mathbb{Z} of the bundle is given by n−mn-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 p 12 2−1=0(mod7)\frac{p_1}{2}^2 -1 = 0 (mod\,7).

By using the connected sum operation, the set of smooth, non-diffeomorphic structures on the nn-sphere has the structure of an abelian group. For the 7-sphere, it is the cyclic group ℤ/28\mathbb{Z}/{28} and Brieskorn (1966) found the generator Σ\Sigma so that Σ#⋯#Σ⏟ 28\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 XX with (exotic) 7-sphere boundaries in Milnor’s construction correspond to near horizon limits of black M2 brane spacetimes ℝ 2,1×X\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 3S^3-bundles over S 4S^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)