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fiber space, space attachment
Extra stuff, structure, properties
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Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
For $n \in \mathbb{N}$ write $U(n)$ for the unitary group in dimension $n$ and $O(n)$ for the orthogonal group in dimension $n$, both regarded as topological groups in the standard way. Write $B U(n)$, $B O(n)$ $\in$ Top for the corresponding classifying space.
Write
and
for the set of homotopy-classes of continuous functions $X \to B U(n)$.
This is equivalently the set of isomorphism classes of $O(n)$- or $U(n)$-principal bundles on $X$ as well as of rank-$n$ real or complex vector bundles on $X$, respectively:
For each $n$ let
be the inclusion of topological groups given by inclusion of $n \times n$ matrices into $(n+1) \times (n+1)$-matrices given by the block-diagonal form
This induces a corresponding sequence of morphisms of classifying spaces, def. , in Top
Write
for the homotopy colimit (the “homotopy direct limit”) over this diagram (see at homotopy colimit the section Sequential homotopy colimits).
The topological space $B U$ is not equivalent to $B U(\mathcal{H})$, where $U(\mathcal{H})$ is the unitary group on an infinite-dimensional separable Hilbert space $\mathcal{H}$. In fact the latter is contractible, hence has a weak homotopy equivalence to the point
while $B U$ has nontrivial homotopy groups in arbitrary higher degree (by Kuiper's theorem).
But there is the subgroup $U(\mathcal{H})_{\mathcal{K}} \subset U(\mathcal{H})$ of unitary operators that differ from the identity by a compact operator. This is essentially $U = \Omega B U$. See below.
Write $\mathbb{Z}$ for the set of integers regarded as a discrete topological space.
The product spaces
are classifying spaces for real and complex topological K-theory, respectively: for every compact Hausdorff topological space $X$, we have an isomorphism of groups
See for instance (Friedlander, prop. 3.2) or (Karoubi, prop. 1.32, theorem 1.33).
First consider the statement for reduced cohomology $\tilde K(X)$:
Since a compact topological space is a compact object in Top (and using that the classifying spaces $B U(n)$ 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
Since $[X, B U(n)] \simeq U(n) Bund(X)$, 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
Because $H^0(X,\mathbb{Z}) \simeq [X, \mathbb{Z}]$ it follows that
There is another variant on the classifying space
Let
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:
$U_\mathcal{K}$ is a homotopy equivalent model for $B U$. It is in fact the norm closure of the evident model of $B U$ in $U(\mathcal{H})$.
Moreover $U_{\mathcal{K}} \subset U(\mathcal{H})$ is a Banach Lie normal subgroup.
Since $U(\mathcal{H})$ is contractible, it follows that
is a model for the classifying space of reduced complex topological K-theory KU.
Discussion in the context of topological K-theory:
Allen Hatcher, Vector bundles and K-theory (web)
Eric Friedlander, An introduction to K-theory (pdf)
Max Karoubi, K-Theory – An introduction, Grundlehren der mathematischen Wissenschaften 226, Springer 1978 (pdf, doi:10.1007%2F978-3-540-79890-3)
H. Blaine Lawson, Marie-Louise Michelsohn, Spin geometry, Princeton University Press (1989)
On the H-space structure on $B U \times \mathbb{Z}$:
Last revised on March 11, 2024 at 05:25:23. See the history of this page for a list of all contributions to it.