nLab ninebrane 10-group

Contents

Contents

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.

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Whitehead tower of orthogonal grouporientationspin groupstring groupfivebrane groupninebrane group
higher versionsspecial orthogonal groupspin groupstring 2-groupfivebrane 6-groupninebrane 10-group
homotopy groups of stable orthogonal groupπ n(O)\pi_n(O) 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2
stable homotopy groups of spheresπ n(𝕊)\pi_n(\mathbb{S})\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_2 24\mathbb{Z}_{24}00 2\mathbb{Z}_2 240\mathbb{Z}_{240} 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 6\mathbb{Z}_6 504\mathbb{Z}_{504}0 3\mathbb{Z}_3 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 480 2\mathbb{Z}_{480} \oplus \mathbb{Z}_2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2
image of J-homomorphismim(π n(J))im(\pi_n(J))0 2\mathbb{Z}_20 24\mathbb{Z}_{24}000 240\mathbb{Z}_{240} 2\mathbb{Z}_2 2\mathbb{Z}_20 504\mathbb{Z}_{504}000 480\mathbb{Z}_{480} 2\mathbb{Z}_2
smooth ∞-groupWhitehead tower of smooth moduli ∞-stacksG-structure/higher spin structureobstruction
\vdots
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ninebrane 10-groupBNinebrane\mathbf{B}Ninebrane ninebrane structurethird fractional Pontryagin class
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fivebrane 6-groupBFivebrane1np 3B 11U(1)\mathbf{B}Fivebrane \stackrel{\tfrac{1}{n} p_3}{\to} \mathbf{B}^{11}U(1)fivebrane structuresecond fractional Pontryagin class
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string 2-groupBString16p 2B 7U(1)\mathbf{B}String \stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to} \mathbf{B}^7 U(1)string structurefirst fractional Pontryagin class
\downarrow
spin groupBSpin12p 1B 3U(1)\mathbf{B}Spin \stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to} \mathbf{B}^3 U(1)spin structuresecond Stiefel-Whitney class
\downarrow
special orthogonal groupBSOw 2B 2 2\mathbf{B}SO \stackrel{\mathbf{w_2}}{\to} \mathbf{B}^2 \mathbb{Z}_2orientation structurefirst Stiefel-Whitney class
\downarrow
orthogonal groupBOw 1B 2\mathbf{B}O \stackrel{\mathbf{w}_1}{\to} \mathbf{B}\mathbb{Z}_2orthogonal structure/vielbein/Riemannian metric
\downarrow
general linear groupBGL\mathbf{B}GLsmooth manifold

(all hooks are homotopy fiber sequences)

References

Last revised on May 27, 2022 at 02:59:19. See the history of this page for a list of all contributions to it.