nLab BSO(n)




BSO(n)B SO(n) is the classifying space for the special orthogonal group SO(n)SO(n).



BSO(n)B SO(n) is the colimit of the sequence of canonical inclusions of real orientable Grassmannians Gr˜ n( k)Gr˜ n( k+1)\widetilde{Gr}_n(\mathbb{R}^k)\hookrightarrow\widetilde{Gr}_n(\mathbb{R}^{k+1}):

BSO(n)lim nGr˜ n( k) B SO(n) \coloneqq\underset{\longrightarrow}{\lim}_n\widetilde{Gr}_n(\mathbb{R}^k)

As the real orientable Grassmannian can be written as a homogenous space by

Gr˜ n( k)=SO(k)/(SO(n)×SO(kn)) \widetilde{Gr}_n(\mathbb{R}^k) =SO(k)/(SO(n)\times SO(k-n))

the group structure carries over to BSO(n)B SO(n).

Smallest classifying spaces

Since SO(1)1SO(1)\cong 1 is the trivial group, the classifying space BSO(1)B SO(1) is the trivial topological space. Since SO(2)U(1)SO(2)\cong U(1), once has

BSO(2)BU(1)P . B SO(2) \cong B U(1) \cong\mathbb{C}P^\infty.



The cohomology ring of BSO(n)B S O(n) with coefficients in the field 2\mathbb{Z}_2 is generated by the Stiefel-Whitney classes and given by

H *(BSO(n); 2) 2[w 2,,w n]. H^*(B SO(n);\mathbb{Z}_2) \cong\mathbb{Z}_2[w_2,\ldots,w_n].

(Milnor & Stasheff 74, Theorem 12.4.), (Hatcher 02, Example 4D.6.)

This result holds more generally for every field with characteristic char=2\operatorname{char}=2.


The cohomology ring of BSO(n)B SO(n) with coefficients in the field \mathbb{Q} of rational numbers is generated by the Pontrjagin classes as well as the Euler class and given by

H *(BSO(2n);)[p 1,,p n,e]/(p ne 2) H^*(B SO(2n);\mathbb{Q}) \cong\mathbb{Q}[p_1,\ldots,p_n,e]/(p_n-e^2)
H *(BSO(2n+1);)[p 1,,p n] H^*(B SO(2n+1);\mathbb{Q}) \cong\mathbb{Q}[p_1,\ldots,p_n]

These results hold more generally for every field with characteristic char2\operatorname{char}\neq 2.


The canonical inclusions SO(n)SO(n+1)SO(n)\hookrightarrow SO(n+1) yield canonical inclusions BSO(n)BSO(n+1)B SO(n)\hookrightarrow B SO(n+1) of their respective classifying spaces. The colimit is denoted as

BSOlim nBSO(n). B SO \coloneqq\underset{\longrightarrow}{\lim}_n B SO(n).

and indeed the classifying space for SOlim nSO(n)SO\coloneqq\underset{\longrightarrow}{\lim}_n SO(n).


Created on March 14, 2024 at 15:50:24. See the history of this page for a list of all contributions to it.