- Source: Classifying space for O(n)
In mathematics, the classifying space for the orthogonal group O(n) may be constructed as the Grassmannian of n-planes in an infinite-dimensional real space
R
∞
{\displaystyle \mathbb {R} ^{\infty }}
.
Cohomology ring
The cohomology ring of
BO
(
n
)
{\displaystyle \operatorname {BO} (n)}
with coefficients in the field
Z
2
{\displaystyle \mathbb {Z} _{2}}
of two elements is generated by the Stiefel–Whitney classes:
H
∗
(
BO
(
n
)
;
Z
2
)
=
Z
2
[
w
1
,
…
,
w
n
]
.
{\displaystyle H^{*}(\operatorname {BO} (n);\mathbb {Z} _{2})=\mathbb {Z} _{2}[w_{1},\ldots ,w_{n}].}
Infinite classifying space
The canonical inclusions
O
(
n
)
↪
O
(
n
+
1
)
{\displaystyle \operatorname {O} (n)\hookrightarrow \operatorname {O} (n+1)}
induce canonical inclusions
BO
(
n
)
↪
BO
(
n
+
1
)
{\displaystyle \operatorname {BO} (n)\hookrightarrow \operatorname {BO} (n+1)}
on their respective classifying spaces. Their respective colimits are denoted as:
O
:=
lim
n
→
∞
O
(
n
)
;
{\displaystyle \operatorname {O} :=\lim _{n\rightarrow \infty }\operatorname {O} (n);}
BO
:=
lim
n
→
∞
BO
(
n
)
.
{\displaystyle \operatorname {BO} :=\lim _{n\rightarrow \infty }\operatorname {BO} (n).}
BO
{\displaystyle \operatorname {BO} }
is indeed the classifying space of
O
{\displaystyle \operatorname {O} }
.
See also
Classifying space for U(n)
Classifying space for SO(n)
Classifying space for SU(n)
Literature
Milnor, John; Stasheff, James (1974). Characteristic classes (PDF). Princeton University Press. doi:10.1515/9781400881826. ISBN 9780691081229.
Hatcher, Allen (2002). Algebraic topology. Cambridge: Cambridge University Press. ISBN 0-521-79160-X.
Mitchell, Stephen (August 2001). Universal principal bundles and classifying spaces (PDF).{{cite book}}: CS1 maint: year (link)
External links
classifying space on nLab
BO(n) on nLab