nLab 16-manifold

Idea

A 16-manifold is a manifold of dimension 16. (Generally a topological manifold, but it can be specified to a PL manifold or a smooth manifold.)

Examples

• 16-sphere
• Unique exotic 16-sphere (the Kervaire-Milnor group is $\Theta_16\cong\mathbb{Z}_2$)
• real projective space$\mathbb{R}P^16$
• complex projective space$\mathbb{C}P^8$
• complex projective space$\mathbb{H}P^4$
• U(4)
• Spinᶜ(6)

Properties

Proposition

(Hirzebruch signature theorem) For an orientable smooth 16-manifold $M$ with fundamental class $[M]\in H^16(M,\mathbb{Z})\cong\mathbb{Z}$, its signature is given by Pontrjagin numbers as:

$$\sigma(M) =\frac{1}{14175}\langle(381 p_4-71p_1p_3-19p_2^2+22p_1^2p_2-3p_1^4)(M),[M]\rangle \in\mathbb{Z}.$$

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