stable unitary group in nLab

Definition

For n∈ℕn \in \mathbb{N} write U(n)U(n) for the unitary group in dimension nn and O(n)O(n) for the orthogonal group in dimension nn, both regarded as topological groups in the standard way. Write B U ( n ) B U(n) , B O ( n ) B O(n) ∈\in Top for the corresponding classifying space.

Write

[X,BO(n)]:=π 0Top(X,BO(n)) [X, B O(n)] := \pi_0 Top(X, B O(n))

and

[X,BU(n)]:=π 0Top(X,BU(n)) [X, B U(n)] := \pi_0 Top(X, B U(n))

for the set of homotopy-classes of continuous functions X→BU(n)X \to B U(n).

Proposition

This is equivalently the set of isomorphism classes of O(n)O(n)- or U(n)U(n)-principal bundles on XX as well as of rank-nn real or complex vector bundles on XX, respectively:

[X,BO(n)]≃O(n)Bund(X)≃Vect ℝ(X,n), [X, B O(n)] \simeq O(n) Bund(X) \simeq Vect_{\mathbb{R}}(X,n) \,,

[X,BU(n)]≃U(n)Bund(X)≃Vect ℂ(X,n). [X, B U(n)] \simeq U(n) Bund(X) \simeq Vect_{\mathbb{C}}(X,n) \,.

Definition

For each nn let

U(n)→U(n+1) U(n) \to U(n+1)

be the inclusion of topological groups given by inclusion of n×nn \times n matrices into (n+1)×(n+1)(n+1) \times (n+1)-matrices given by the block-diagonal form

[g]↦[1 [0] [0] [g]]. \left[g\right] \mapsto \left[ \array{ 1 & [0] \\ [0] & [g] } \right] \,.

This induces a corresponding sequence of morphisms of classifying spaces, def. , in Top

BU(0)↪BU(1)↪BU(2)↪⋯. B U(0) \hookrightarrow B U(1) \hookrightarrow B U(2) \hookrightarrow \cdots \,.

Write

BU≔lim → n∈ℕBU(n) B U \;\coloneqq\; {\lim_{\to}}_{n \in \mathbb{N}} B U(n)

for the homotopy colimit (the “homotopy direct limit”) over this diagram (see at homotopy colimit the section Sequential homotopy colimits).

Proposition

Write ℤ\mathbb{Z} for the set of integers regarded as a discrete topological space.

The product spaces

BO×ℤ,BU×ℤ B O \times \mathbb{Z}\,,\;\;\;\;\;B U \times \mathbb{Z}

are classifying spaces for real and complex topological K-theory, respectively: for every compact Hausdorff topological space XX, we have an isomorphism of groups

K˜(X)≃[X,BU]. \tilde K(X) \simeq [X, B U ] \,.

K(X)≃[X,BU×ℤ]. K(X) \simeq [X, B U \times \mathbb{Z}] \,.

Proof

First consider the statement for reduced cohomology K˜(X)\tilde K(X):

Since a compact topological space is a compact object in Top (and using that the classifying spaces BU(n)B U(n) are (see there) paracompact topological spaces, hence normal, and since the inclusion morphisms are closed inclusions (…)) the hom-functor out of it commutes with the filtered colimit

Top(X,BU) =Top(X,lim → nBU(n)) ≃lim → nTop(X,BU(n)). \begin{aligned} Top(X, B U) &= Top(X, {\lim_\to}_n B U(n)) \\ & \simeq {\lim_\to}_n Top(X, B U (n)) \end{aligned} \,.

Since [X,BU(n)]≃U(n)Bund(X)[X, B U(n)] \simeq U(n) Bund(X), in the last line the colimit is over vector bundles of all ranks and identifies two if they become isomorphic after adding a trivial bundle of some finite rank.

For the full statement use that by prop. we have

K(X)≃H 0(X,ℤ)⊕K˜(X). K(X) \simeq H^0(X, \mathbb{Z}) \oplus \tilde K(X) \,.

Because H 0(X,ℤ)≃[X,ℤ]H^0(X,\mathbb{Z}) \simeq [X, \mathbb{Z}] it follows that

H 0(X,ℤ)⊕K˜(X)≃[X,ℤ]×[X,BU]≃[X,BU×ℤ]. H^0(X, \mathbb{Z}) \oplus \tilde K(X) \simeq [X, \mathbb{Z}] \times [X, B U] \simeq [X, B U \times \mathbb{Z}] \,.