nLab 7-manifold

Idea

An 7-manifold is a manifold of dimension 7. (Generally a topological manifold, but it can be specified to a PL manifold or a smooth manifold.) 7-manifolds are of particular interest in the compactification, mainly Freund-Rubin compactification, of theories on 11-manifolds, mainly M-theory and its low energy limit in D=11 supergravity, to the 4-manifold observed for spacetime. It’s also the lowest dimension in which the phenomenon of exotic spheres arises. G₂ manifolds are special 7-manifolds.

Examples

• 7-sphere, which appears as total space in the quaternionic Hopf fibration $h_\mathbb{H}\colon S^7\twoheadrightarrow S^4$ and the quaternionic Arnold-Kuiper-Massey theorem $\mathbb{H}P^2/U(1)\cong S^7$. It is also given by the exceptional diffeomorphisms $S^7\cong Spin(5)/SU(2)\cong Spin(6)/SU(3)\cong Spin(7)/G_2$, see Spin(5)/SU(2) is the 7-sphere and Spin(7)/G₂ is the 7-sphere.
• exotic 7-sphere, including the Gromoll-Meyer sphere
• real projective space$\mathbb{R}P^7$
• Spinᶜ(4)

Properties

Proposition

Every orientable 7-manifold bounds a orientable 8-manifold.

Equivalently, the seventh oriented bordism group is trivial, which is the case as can be computed by the seventh homotopy group of the special orthogonal Thom spectrum MSO according to Thom's theorem:

$$\Omega_7^SO =\pi_7 MSO \cong 1.$$

It is the last trivial oriented bordism group.

Proposition

For a smooth 7-manifold $M$, its last three Stiefel-Whitney classes vanish:

$$w_5(M) =w_6(M) =w_7(M) =0.$$

More generally, for every $4n+3$-dimensional smooth manifold, one has $w_{4n+1}=w_{4n+2}(M)=w_{4n+3}(M)=0$. Hence this property also holds for 3-manifolds, 11-manifolds and 15-manifolds.

Proposition

If for a smooth 15-manifold $M$ the following Stiefel-Whitney classes vanish:

$$w_1(M)=w_2(M)=w_4(M)=0,$$

then all its Stiefel-Whitney classes vanish.

Related concepts

manifolds in low dimension:

• 2-manifolds

• 3-manifolds

• 4-manifolds

• 5-manifolds

• 6-manifolds

• 7-manifolds

• 8-manifolds

• 9-manifolds

• 10-manifolds

• 11-manifolds

• 12-manifolds

• 13-manifolds

• 14-manifolds

• 15-manifolds

• 16-manifolds

Applications of 7-manifolds:

• D=7 Chern-Simons theory
• D=7 Seiberg-Witten theory
• D=7 super Yang-Mills theory
• D=7 supergravity

References

On exotic 7-spheres:

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