nLab 2-orient group

Idea

A 2-Orient group is a topological group appearing in the Whitehead tower of the orthogonal group as an intermediate step between the more important fivebrane group and ninebrane group:

$$\ldots \rightarrow Ninebrane(n) \rightarrow 2\text{-}Spin(n) \rightarrow 2\text{-}Orient(n) \rightarrow Fivebrane(n) \rightarrow String(n) \rightarrow Spin(n) \rightarrow SO(n) \rightarrow O(n).$$

2-Orient groups are denoted that way, because in limit the construction removes the same homotopy group as for the special orthogonal group, which describes orientation, under the eight-fold Bott periodicity:

$$2\text{-}Orient \coloneqq O\langle 9\rangle;$$

$$SO =O\langle 1\rangle.$$

(In similar fashion, ninebrane groups would be the 2-string groups.)

References

• Hisham Sati, Ninebrane structures (arXiv:1405.7686)