Classifying space for SU(n) explained

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) → \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} ^{k +1} _{n( C} ), V\mapsto V x \{ 0 \} . Its colimit is:\operatorname{BSU}(n) : = \widetilde\operatorname{Gr} ^{inf ty) : = \lim} _{n → inf ty} \widetilde\operatorname{Gr} ^{k).} _{n( C}

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

\widetilde\operatorname{Gr} ^{k) = \operatorname{SU}(n + k)/(\operatorname{SU}(n) x \operatorname{SU}(k))} _{n( C}

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 H P ^{infty} .

Classification of principal bundles

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:^[1] [X,\operatorname{BSU}(n)] → \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 Z of integers is generated by the Chern classes:^[2] H ^{*(\operatorname{BSU}(n); Z ) = 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 → infty} \operatorname{SU}(n); \operatorname{BSU} : = \lim _{n → infty} \operatorname{BSU}(n). \operatorname{BSU} is indeed the classifying space of \operatorname{SU} .

See also

• Classifying space for O(n)
• Classifying space for SO(n)
• Classifying space for U(n)

Literature

• Book: Hatcher, Allen. Algebraic topology. Cambridge University Press. Cambridge. 2002. 0-521-79160-X.
• Book: Mitchell, Stephen. Universal principal bundles and classifying spaces. August 2001.

External links

• classifying space on nLab
• BSU(n) on nLab

References

1. Web site: 2024-03-14 . en . universal principal bundle . nLab.
2. Hatcher 02, Example 4D.7.