Classifying space for SU(n)#Infinite classifying space

In mathematics, the classifying space $\operatorname{BSU}(n)$ for the special unitary group $\operatorname{SU}(n)$ is the base space of the universal $\operatorname{SU}(n)$ principal bundle $\operatorname{ESU}(n)\rightarrow\operatorname{BSU}(n)$ . This means that $\operatorname{SU}(n)$ principal bundles over a CW complex up to isomorphism are in bijection with homotopy classes of its continuous maps into $\operatorname{BSU}(n)$ . The isomorphism is given by pullback.

Definition

There is a canonical inclusion of complex oriented Grassmannians given by $\widetilde\operatorname{Gr}_n(\mathbb{C}^k)\hookrightarrow\widetilde\operatorname{Gr}_n(\mathbb{C}^{k+1}),
V\mapsto V\times\{0\}$ . Its colimit is:

$\operatorname{BSU}(n)
:=\widetilde\operatorname{Gr}_n(\mathbb{C}^\infty)
:=\lim_{n\rightarrow\infty}\widetilde\operatorname{Gr}_n(\mathbb{C}^k).$

Since real oriented Grassmannians can be expressed as a homogeneous space by:

: $\widetilde\operatorname{Gr}_n(\mathbb{C}^k)
=\operatorname{SU}(n+k)/(\operatorname{SU}(n)\times\operatorname{SU}(k))$

the group structure carries over to $\operatorname{BSU}(n)$ .

Simplest classifying spaces

Since $\operatorname{SU}(1)$

\cong 1 is the trivial group,

\operatorname{BSU}(1)
\cong\{*\}

is the trivial topological space.

Since $\operatorname{SU}(2)$

\cong\operatorname{Sp}(1), one has

\operatorname{BSU}(2)
\cong\operatorname{BSp}(1)
\cong\mathbb{H}P^\infty

Classification of principal bundles

Given a topological space $X$ the set of $\operatorname{SU}(n)$ principal bundles on it up to isomorphism is denoted $\operatorname{Prin}_{\operatorname{SU}(n)}(X)$ . If $X$ is a CW complex, then the map:{{cite web |access-date=2024-03-14 |language=en |title=universal principal bundle |url=https://ncatlab.org/nlab/show/universal+principal+bundle |website=nLab}}

: $[X,\operatorname{BSU}(n)]\rightarrow\operatorname{Prin}_{\operatorname{SU}(n)}(X),
[f]\mapsto f^*\operatorname{ESU}(n)$

is bijective.

Cohomology ring

The cohomology ring of $\operatorname{BSU}(n)$ with coefficients in the ring $\mathbb{Z}$ of integers is generated by the Chern classes:Hatcher 02, Example 4D.7.

: $H^*(\operatorname{BSU}(n);\mathbb{Z})
=\mathbb{Z}[c_2,\ldots,c_n].$

Infinite classifying space

The canonical inclusions $\operatorname{SU}(n)\hookrightarrow\operatorname{SU}(n+1)$ induce canonical inclusions $\operatorname{BSU}(n)\hookrightarrow\operatorname{BSU}(n+1)$ on their respective classifying spaces. Their respective colimits are denoted as:

: $\operatorname{SU}
:=\lim_{n\rightarrow\infty}\operatorname{SU}(n);$

: $\operatorname{BSU}
:=\lim_{n\rightarrow\infty}\operatorname{BSU}(n).$

$\operatorname{BSU}$ is indeed the classifying space of $\operatorname{SU}$ .

Literature

{{cite book|last=Hatcher|first=Allen|title=Algebraic topology|publisher=Cambridge University Press|location=Cambridge|year=2002|isbn=0-521-79160-X|url=https://pi.math.cornell.edu/~hatcher/AT/ATpage.html}}
{{cite book|title=Universal principal bundles and classifying spaces|publisher=|location=|year=August 2001|isbn=|url=https://math.mit.edu/~mbehrens/18.906/prin.pdf|last=Mitchell|first=Stephen|doi=}}

External links

classifying space on nLab
BSU(n) on nLab

References

Category:Algebraic topology