nLab 15-manifold

Idea

An 15-manifold is a manifold of dimension 15. (Generally a topological manifold, but it can be specified to a PL manifold or a smooth manifold.)

Examples

• 15-sphere
• real projective space$\mathbb{R}P^15$
• O(6)
• SO(6)
• Spin(6)$\cong$SU(4)

Properties

Proposition

For a smooth 15-manifold $M$, its last three Stiefel-Whitney classes vanish:

$$w_13(M) =w_14(M) =w_15(M) =0.$$

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, 7-manifolds and 11-manifolds.

Proposition

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

$$w_1(M)=w_2(M)=w_4(M)=w_8(M)=0,$$

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