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Group Theory




For 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 BU(n),BO(n)B U(n) , B O(n) \in Top for the corresponding classifying space.


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


[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 XBU(n)X \to B U(n).


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

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 \,.


BU:=lim nBU(n) B U := {\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).


The topological space BUB U is not equivalent to BU()B U(\mathcal{H}) , where U()U(\mathcal{H}) is the unitary group on a separable infinite-dimensional Hilbert space \mathcal{H}. In fact the latter is contractible, hence has a weak homotopy equivalence to the point

BU()* B U(\mathcal{H}) \simeq *

while BUB U has nontrivial homotopy groups in arbitrary higher degree (by Kuiper's theorem).

But there is the group U() 𝒦U()U(\mathcal{H})_{\mathcal{K}} \subset U(\mathcal{H}) of unitary operators that differ from the identity by a compact operator. This is essentially U=ΩBUU = \Omega B U. See below.


Classifying space for topological K-theory


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

The product spaces

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

See for instance (Friedlander, prop. 3.2) or (Karoubi, prop. 1.32, theorem 1.33).


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

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

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

There is another variant on the classifying space



U 𝒦={gU()|gid𝒦} U_{\mathcal{K}} = \left\{ g \in U(\mathcal{H}) | g - id \in \mathcal{K} \right\}

be the group of unitary operators on a separable Hilbert space \mathcal{H} which differ from the identity by a compact operator.

Palais showed that


U 𝒦U_\mathcal{K} is a homotopy equivalent model for BUB U. It is in fact the norm closure of the evident model of BUB U in U()U(\mathcal{H}).

Moreover U 𝒦U()U_{\mathcal{K}} \subset U(\mathcal{H}) is a Banach Lie normal subgroup.

Since U()U(\mathcal{H}) is contractible, it follows that

BU 𝒦U()/U 𝒦 B U_{\mathcal{K}} \coloneqq U(\mathcal{H})/U_{\mathcal{K}}

is a model for the classifying space of reduced complex topological K-theory KU.


Discussion in the context of topological K-theory:

Discussion in the broader context of twisted equivariant topological K-theory:

On the H-space structure on BU×B U \times \mathbb{Z}:

Last revised on June 18, 2021 at 12:47:26. See the history of this page for a list of all contributions to it.