- Source: Classifying space for SU(n)
In mathematics, the classifying space
BSU
(
n
)
{\displaystyle \operatorname {BSU} (n)}
for the special unitary group
SU
(
n
)
{\displaystyle \operatorname {SU} (n)}
is the base space of the universal
SU
(
n
)
{\displaystyle \operatorname {SU} (n)}
principal bundle
ESU
(
n
)
→
BSU
(
n
)
{\displaystyle \operatorname {ESU} (n)\rightarrow \operatorname {BSU} (n)}
. This means that
SU
(
n
)
{\displaystyle \operatorname {SU} (n)}
principal bundles over a CW complex up to isomorphism are in bijection with homotopy classes of its continuous maps into
BSU
(
n
)
{\displaystyle \operatorname {BSU} (n)}
. The isomorphism is given by pullback.
Definition
There is a canonical inclusion of complex oriented Grassmannians given by
Gr
~
n
(
C
k
)
↪
Gr
~
n
(
C
k
+
1
)
,
V
↦
V
×
{
0
}
{\displaystyle {\widetilde {\operatorname {Gr} }}_{n}(\mathbb {C} ^{k})\hookrightarrow {\widetilde {\operatorname {Gr} }}_{n}(\mathbb {C} ^{k+1}),V\mapsto V\times \{0\}}
. Its colimit is:
BSU
(
n
)
:=
Gr
~
n
(
C
∞
)
:=
lim
n
→
∞
Gr
~
n
(
C
k
)
.
{\displaystyle \operatorname {BSU} (n):={\widetilde {\operatorname {Gr} }}_{n}(\mathbb {C} ^{\infty }):=\lim _{n\rightarrow \infty }{\widetilde {\operatorname {Gr} }}_{n}(\mathbb {C} ^{k}).}
Since real oriented Grassmannians can be expressed as a homogeneous space by:
Gr
~
n
(
C
k
)
=
SU
(
n
+
k
)
/
(
SU
(
n
)
×
SU
(
k
)
)
{\displaystyle {\widetilde {\operatorname {Gr} }}_{n}(\mathbb {C} ^{k})=\operatorname {SU} (n+k)/(\operatorname {SU} (n)\times \operatorname {SU} (k))}
the group structure carries over to
BSU
(
n
)
{\displaystyle \operatorname {BSU} (n)}
.
Simplest classifying spaces
Since
SU
(
1
)
≅
1
{\displaystyle \operatorname {SU} (1)\cong 1}
is the trivial group,
BSU
(
1
)
≅
{
∗
}
{\displaystyle \operatorname {BSU} (1)\cong \{*\}}
is the trivial topological space.
Since
SU
(
2
)
≅
Sp
(
1
)
{\displaystyle \operatorname {SU} (2)\cong \operatorname {Sp} (1)}
, one has
BSU
(
2
)
≅
BSp
(
1
)
≅
H
P
∞
{\displaystyle \operatorname {BSU} (2)\cong \operatorname {BSp} (1)\cong \mathbb {H} P^{\infty }}
.
Classification of principal bundles
Given a topological space
X
{\displaystyle X}
the set of
SU
(
n
)
{\displaystyle \operatorname {SU} (n)}
principal bundles on it up to isomorphism is denoted
Prin
SU
(
n
)
(
X
)
{\displaystyle \operatorname {Prin} _{\operatorname {SU} (n)}(X)}
. If
X
{\displaystyle X}
is a CW complex, then the map:
[
X
,
BSU
(
n
)
]
→
Prin
SU
(
n
)
(
X
)
,
[
f
]
↦
f
∗
ESU
(
n
)
{\displaystyle [X,\operatorname {BSU} (n)]\rightarrow \operatorname {Prin} _{\operatorname {SU} (n)}(X),[f]\mapsto f^{*}\operatorname {ESU} (n)}
is bijective.
Cohomology ring
The cohomology ring of
BSU
(
n
)
{\displaystyle \operatorname {BSU} (n)}
with coefficients in the ring
Z
{\displaystyle \mathbb {Z} }
of integers is generated by the Chern classes:
H
∗
(
BSU
(
n
)
;
Z
)
=
Z
[
c
2
,
…
,
c
n
]
.
{\displaystyle H^{*}(\operatorname {BSU} (n);\mathbb {Z} )=\mathbb {Z} [c_{2},\ldots ,c_{n}].}
Infinite classifying space
The canonical inclusions
SU
(
n
)
↪
SU
(
n
+
1
)
{\displaystyle \operatorname {SU} (n)\hookrightarrow \operatorname {SU} (n+1)}
induce canonical inclusions
BSU
(
n
)
↪
BSU
(
n
+
1
)
{\displaystyle \operatorname {BSU} (n)\hookrightarrow \operatorname {BSU} (n+1)}
on their respective classifying spaces. Their respective colimits are denoted as:
SU
:=
lim
n
→
∞
SU
(
n
)
;
{\displaystyle \operatorname {SU} :=\lim _{n\rightarrow \infty }\operatorname {SU} (n);}
BSU
:=
lim
n
→
∞
BSU
(
n
)
.
{\displaystyle \operatorname {BSU} :=\lim _{n\rightarrow \infty }\operatorname {BSU} (n).}
BSU
{\displaystyle \operatorname {BSU} }
is indeed the classifying space of
SU
{\displaystyle \operatorname {SU} }
.
See also
Classifying space for O(n)
Classifying space for SO(n)
Classifying space for U(n)
Literature
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
BSU(n) on nLab