nLab ninebrane 10-group

Idea

The 10-group which is the looping of the homotopy fiber of (some fractional multiple of) the third Pontryagin class on the delooping of the fivebrane 6-group.

Related concepts

• M9-brane

	$n$	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Whitehead tower of orthogonal group		orientation	spin group		string group				fivebrane group				ninebrane group
higher versions		special orthogonal group	spin group		string 2-group				fivebrane 6-group				ninebrane 10-group
homotopy groups of stable orthogonal group	$\pi_n(O)$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$	$\mathbb{Z}_2$
stable homotopy groups of spheres	$\pi_n(\mathbb{S})$	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z}_{24}$	0	0	$\mathbb{Z}_2$	$\mathbb{Z}_{240}$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_6$	$\mathbb{Z}_{504}$	0	$\mathbb{Z}_3$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_{480} \oplus \mathbb{Z}_2$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$
image of J-homomorphism	$im(\pi_n(J))$	0	$\mathbb{Z}_2$	0	$\mathbb{Z}_{24}$	0	0	0	$\mathbb{Z}_{240}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}_{504}$	0	0	0	$\mathbb{Z}_{480}$	$\mathbb{Z}_2$

smooth ∞-group	Whitehead tower of smooth moduli ∞-stacks	G-structure/higher spin structure	obstruction
	$\vdots$
	$\downarrow$
ninebrane 10-group	$\mathbf{B}Ninebrane$	ninebrane structure	third fractional Pontryagin class
	$\downarrow$
fivebrane 6-group	$\mathbf{B}Fivebrane \stackrel{\tfrac{1}{n} p_3}{\to} \mathbf{B}^{11}U(1)$	fivebrane structure	second fractional Pontryagin class
	$\downarrow$
string 2-group	$\mathbf{B}String \stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to} \mathbf{B}^7 U(1)$	string structure	first fractional Pontryagin class
	$\downarrow$
spin group	$\mathbf{B}Spin \stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to} \mathbf{B}^3 U(1)$	spin structure	second Stiefel-Whitney class
	$\downarrow$
special orthogonal group	$\mathbf{B}SO \stackrel{\mathbf{w_2}}{\to} \mathbf{B}^2 \mathbb{Z}_2$	orientation structure	first Stiefel-Whitney class
	$\downarrow$
orthogonal group	$\mathbf{B}O \stackrel{\mathbf{w}_1}{\to} \mathbf{B}\mathbb{Z}_2$	orthogonal structure/vielbein/Riemannian metric
	$\downarrow$
general linear group	$\mathbf{B}GL$	smooth manifold

(all hooks are homotopy fiber sequences)

References

• Hisham Sati, Ninebrane structures (arXiv:1405.7686)