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Results tagged with classifying-spaces
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user 8032
The classifying space BG of a group G classifies principal G-bundles, in that homotopy classes of maps [X, BG] are naturally identified with isomorphism classes of principal G-bundles P ⭢ X.
8
votes
what does BG classify? i.e. what is a principal fibration?
When $G$ is discrete, another sort of answer is provided by the paper of Michael Weiss:
What does the classifying space of a category classify? Homology, Homotopy and Applications 7 (2005), 185–195.
…
- 18.5k
10
votes
Dualizable classifying spaces
Dualizable is equivalent to $BG$ being finitely dominated (a retract up to homotopy of a finite cell complex). In Wall's "Finiteness conditions on CW complexes, I" Wall shows that finite domination i …
- 18.5k
17
votes
Accepted
The classifying space of a gauge group
Proof of (1):
(a). Suppose $X$ and $Y$ are $G$-spaces, the action of $G$ on $X$ is free, and $X\to X/G$ is
a principal bundle, then the space of $G$-equivariant maps
$$
F(X,Y)^G
$$
is the same th …
- 18.5k
7
votes
Is every connected space equivalent to some B(Aut(X))?
If you are willing to relax your wish from having answer of the form $\text{Aut}(X)$
to the more general group-like
topological monoid $\text{Aut}_B(E)$ for a suitable fibration $E \to B$, then the a …
- 18.5k
14
votes
Accepted
Characteristic classes for block bundles
I don't know where the results are written down in one place (perhaps in the book of Madsen and Milgram?), but see the the end of this post for a list of references.
In any case, here is a proof of …
- 18.5k
8
votes
contractible configuration spaces
I believe each of these arguments will work.
Argument 1: Consider $S^n \subset \Bbb R^{n+1} \subset S^{n+1}$, where the last inclusion is
given by the upper hemisphere (which is homeomorphic to $\Bbb …
- 18.5k
10
votes
classifying space of orthogonal groups
$BO$ can be defined as the colimit over $(k,n)$ of Grassmanians $G_k(\Bbb R^n)$ of $k$-dimensional linear subspaces of $\Bbb R^n$ (the limit over $n$ is defined by standard inclusions $\Bbb R^n \subse …
- 18.5k