From Wikipedia, the free encyclopedia
In mathematics, the classifying space
for the special unitary group
is the base space of the universal
principal bundle
. This means that
principal bundles over a CW complex up to isomorphism are in bijection with homotopy classes of its continuous maps into
. The isomorphism is given by pullback.
There is a canonical inclusion of complex oriented Grassmannians given by
. Its colimit is:
Since real oriented Grassmannians can be expressed as a homogeneous space by:
![{\displaystyle {\widetilde {\operatorname {Gr} }}_{n}(\mathbb {C} ^{k})=\operatorname {SU} (n+k)/(\operatorname {SU} (n)\times \operatorname {SU} (k))}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QaNeNnDvEo2e5oNGOyqvCaDm4a2a0ngrDaNs3ajFAzjmPzDKNygdF)
the group structure carries over to
.
Simplest classifying spaces
[edit]
- Since
is the trivial group,
is the trivial topological space.
- Since
, one has
.
Classification of principal bundles
[edit]
Given a topological space
the set of
principal bundles on it up to isomorphism is denoted
. If
is a CW complex, then the map:[1]
![{\displaystyle [X,\operatorname {BSU} (n)]\rightarrow \operatorname {Prin} _{\operatorname {SU} (n)}(X),[f]\mapsto f^{*}\operatorname {ESU} (n)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85othCntCQyjo4nqsOygwNnAnByjrCnjvByjsPaAsQoAi4ats2oDm4)
is bijective.
The cohomology ring of
with coefficients in the ring
of integers is generated by the Chern classes:[2]
![{\displaystyle H^{*}(\operatorname {BSU} (n);\mathbb {Z} )=\mathbb {Z} [c_{2},\ldots ,c_{n}].}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NoNaNzNdCots2ytm0atCNyjnEa2dEnAw3aAo5aNsQzAi1aDs1zjlA)
Infinite classifying space
[edit]
The canonical inclusions
induce canonical inclusions
on their respective classifying spaces. Their respective colimits are denoted as:
![{\displaystyle \operatorname {SU} :=\lim _{n\rightarrow \infty }\operatorname {SU} (n);}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81ztC5yjw5yghEngnFa2wQotrAaNsNnjs3njw3ytoQzqsOntFAoqs1)
![{\displaystyle \operatorname {BSU} :=\lim _{n\rightarrow \infty }\operatorname {BSU} (n).}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FztaQyji3ntaNnDsQoDlAzjKPzAi5oteNoAnAatlFoDeOaNaQajK3)
is indeed the classifying space of
.