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 $(n,m)$.

Let $E_{m,n}\twoheadrightarrow S^4$ denote the real vector bundle corresponding to the integer pair $m,n\in\mathbb{Z}$. There is an orientation-reversing vector bundle isomorphism $E_{m,n}\cong E_{-m,-n}$. (McEnroe 15, Lem. 6.14) Let $\alpha\in H^4(S^4)\cong\mathbb{Z}$ be a generator, then its Euler class and fractional first Pontryagin class are:

$$e(E_{m,n}) =-(m+n)\alpha;$$

$$\frac{p_1(E_{m,n})}{2} =(n-m)\alpha.$$

(McEnroe 15, Eq. (6.17) & Eq. (6.29))

With the choice of a Riemannian metric on it, one has a disk bundle $D(E_{m,n})\twoheadrightarrow S^4$ by taking the disks $D^4\subset\mathbb{R}^4$ of each fiber, a sphere bundle $S(E_{m,n})=\partial D(E_{m,n})\twoheadrightarrow S^4$ by taking the spheres $S^3\subset\mathbb{R}^4$ of each fiber as well as a Thom space $Th(E_{m,n})=D(E_{m,n})/S(E_{m,n})$. None of their topological or smooth structures depend on the choice of the Riemannian metric.

A special case is the quaternionic Hopf fibration $h_\mathbb{H}\colon\mathbb{H}^2\supset S^7\twoheadrightarrow S^4\cong\mathbb{H}P^1,[q,p]\mapsto[q:p]$, which is a principal SU(2)-bundle and the sphere bundle of the real vector bundle $(\gamma_\mathbb{H}^{1,1})_\mathbb{R}\twoheadrightarrow S^4$ represented by the continuous map:

$$S^4 \cong\mathbb{H}P^1 \hookrightarrow\mathbb{H}P^\infty \cong B SU(2) \hookrightarrow B SO(4),$$

which corresponds to $m=0$ and $n=1$. (McEnroe 15, p. 11)

More generally for $n+m=1$, a Morse function with exactly two critical points on $S(E_{m,n})$ exists, implying that it is homeomorphic to $S^7$ according to the Reeb sphere theorem. If it is further diffeomorphic, then it is possible to glue the disk bundle $D(E_{m,n})$ and the disk $D^8$ together along their common boundary $S(E_{m,n})\cong S^7$ using a pullback to obtain the closed 8-manifold:

$$M_{m,n} =D(E_{m,n})+_{S^7}D^8.$$

Due to $H^4(M_{m,n},\mathbb{Z})\cong\mathbb{Z}$ and $H^8(M_{m,n},\mathbb{Z})\cong\mathbb{Z}$, the intersection form is described by a single integer and hence the signature can only be $\sigma(M_{m,n})=1$ with suitable orientation. According to Hirzebruch's signature theorem, the signature is also given by:

$$\sigma(M_{m,n}) =\frac{1}{45}\langle 7p_2(M_{m,n})-p_1^2(M_{m,n}),[M_{m,n}]\rangle$$

(McEnroe 15, Thrm. 6.37)

Since the first Pontrjagin class is known, but the second Pontrjagin class is not, John Milnor eliminated it by projecting to the following modulo-7 diffeomorphism invariant:

$$\begin{aligned} 4(m-n)^2 mod 7 =-\langle p_1^2(M_{m,n}),[M_{m,n}]\rangle mod 7 =45\sigma(M_{m,n}) mod 7 =\pm 45 mod 7 =\pm 3 \\ \Rightarrow \left(\frac{p_1(E_{m,n})}{2}\right)^2 =(m-n)^2 mod 7 =1. \end{aligned}$$

(McEnroe 15, Thrm. 6.44)

$S(E_{m,n})$ with $m+n=1$ is then diffeomorphic to the standard $7$-sphere $S^7$ if and only if this equation holds and hence the modulo-7 diffeomorphism invariant $(m-n)^2-1 mod 7$ vanishes. Examples like $m=1$ and $n=2$ show, that this doesn’t have to be true. In this case, the contradiction in Hirzebruch's signature theorem is resolved by the previously used assumption of $S(E_{m,n})$ and $S^7$ to be diffeomorphic to obtain the glueing $M_{m,n}$ to not be true. $E_{m,n}$ with $m+n=1$ and $(m-n)^2 mod 7\neq 1$ is then homeomorphic but not diffeomorphic to the standard $7$-sphere $S^7$.

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 relation to TQFT:

• Ben Gripaios, Oscar Randal-Williams: TQFTs do not detect the Milnor sphere [arXiv:2601.20828]

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)