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Results tagged with classifying-spaces
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user 5690
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.
6
votes
Intuition for the construction of the space $M_G=EG\times _G M$
In homotopy theory and in homological algebra, there a general idea that for any given operation that you might want to perform, there's a class of "nice" objects for that operation. It is a good idea …
- 43.2k
54
votes
Accepted
How are the classifying space of $E_8$ and $K(\mathbb{Z},4)$ related?
Given a simple Lie group $G$, you can check how far $G$ is from beeing a $K(\mathbb Z,3)$ by looking at the place where the affine vertex gets glued onto the Dynkin diagram, and measuring the length o …
- 43.2k
35
votes
‘Naturally occurring’ $K(\pi, n)$ spaces, for $n \geq 2$
If $M$ is a hyperfinite type $I\!I\!I_1$ factor, then (at least conjecturally), its group of outer automorphisms is a $K(\mathbb Z,3)$.
This is based on the following three properties of that von Neum …
- 43.2k
33
votes
Accepted
What is π_1(BG) for an arbitrary topological group $G$?
If $G$ is homeomorphic to a Cantor set (e.g. $G=\mathbb Z_p$), then $BG$ contains a copy of the Hawaiian earrings in it. To see this, take a sequence of points of $G$ that converges to the identity el …
- 43.2k
10
votes
Accepted
What is the (co-)homology of $K(\mathbb{R}_\delta,n)$?
Given any $\mathbb Q$-vector space $A$ (in your case, $A=\mathbb R_\delta$), the integral homology of $K(A,n)$, which is the same as its rational homology, is given by the cofree graded-cocommutative …
- 43.2k