I recently heard the following fact :


Up to the $15$th skeleton, the classifying space $BE\_8$ and $K(\mathbb{Z},4)$ are homotopy equivalent?


I have two questions on this :


(1) Is there any easy way to see this? Of course, knowing the first fourteen homotopy groups of $E\_8$ is enough but then the question is how does one compute them?


(2) Is there any feasible explanation that suggests that $4$th cohomology classes (possibly related to gerbes), i.e., elements in $H^4(X;\mathbb{Z})$, arise from physical considerations and if $X$ is of dimension $14$ or less then we're classifying $E\_8$-bundles on $X$, thereby suggesting that $E\_8$ arises out of physical considerations?


The last question is a little vague but any pointers would be great!



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 of that tail. For $E\_8$, it's the longest, and so $E\_8$ is the best possible approximation to a $K(\mathbb Z,3)$.


$$\bullet - \bullet - \stackrel{\stackrel{\displaystyle\bullet}|}{\bullet} - \underbrace{\bullet - \bullet - \bullet - \bullet}\_{\text{long tail}} - \circ$$


This is done by labelling the cells of the affine Grassmannian $\Omega G$ by data from the dynlin Diagram, and checking how far you need to go for $\Omega G$ to start looking different than $\mathbb C \mathbb P^\infty$.




---


the affine Grassmannian $\Omega G$ is a very nice space: it's a complex (ind-)variety, and it is stratified by finite dimensional cells. In particular, it has a natural *CW*-decomposition.
Each cells of $\Omega G$ is isomorphic to $\mathbb C^n$, and is in particular of even (real) dimension.


Moreover, $\Omega G$ is a coadjoint orbit of the infinite dimensional Lie group $S^1\ltimes \widetilde {LG}$. Here, the tilde refers to the universal central extension of the loop group $LG$, and $S^1$ acts by reparametrizing the loops.


The inclusion $\Omega G\to Lie(S^1\ltimes \widetilde {LG} )^\* $ can be composed with the projection $$Lie(S^1\ltimes \widetilde {LG})^\* \twoheadrightarrow (\mathfrak t\_{S^1\ltimes \widetilde {LG}})^\* \cong \mathfrak t^\* \oplus \mathbb R \oplus \mathbb R$$ (here $\mathfrak t$ denotes the Lie algebra of the maximal torus $T$ of $G$).
It turns out that the composite lands in a translated copy of $\mathfrak t^\* \oplus \mathbb R$, and so one gets map
$$
\mu:\Omega G \to \mathfrak t^\* \oplus \mathbb R
$$
called the moment map (for the $T\_{S^1\ltimes LG}$ action).


What is important, is that the space $t^\* \oplus \mathbb R$ has a natural basis 
that is indexed by the vertices of the extended Dynkin diagram: those are the *simple coroots*.
I will denote each cell by the moment map image in $\mathfrak t^\* \oplus \mathbb R$ of its center point (in the basis of simple coroots).


Now, let me specialize to the case $G=E\_8$. Here we go:  
  

0-dimensional cell:
$\qquad\begin{matrix}
0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 0 - 0 - 0 - 0
\end{matrix}$  
  

2-dimensional cell:
$\qquad\begin{matrix}
0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 0 - 0 - 0 - 1
\end{matrix}$  
  

4-dimensional cell:
$\qquad\begin{matrix}
0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 0 - 0 - 1 - 1
\end{matrix}$  
  

6-dimensional cell:
$\qquad\begin{matrix}
0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 0 - 1 - 1 - 1
\end{matrix}$  
  

8-dimensional cell:
$\qquad\begin{matrix}
0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 1 - 1 - 1 - 1
\end{matrix}$  
  

10-dimensional cell:
$\qquad\begin{matrix}
0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 1 - 1 - 1 - 1 - 1
\end{matrix}$  
  

12-dimensional cell:
$\qquad\begin{matrix}
0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{1} - 1 - 1 - 1 - 1 - 1
\end{matrix}$  
  

14-dimensional cell:
$\qquad\begin{matrix}
0 - 0 - \stackrel{\stackrel{\displaystyle 1}|}{1} - 1 - 1 - 1 - 1 - 1
\end{matrix}$  
  

other 14-dimensional cell:
$\qquad\begin{matrix}
0 - 1 - \stackrel{\stackrel{\displaystyle 0}|}{1} - 1 - 1 - 1 - 1 - 1
\end{matrix}$  



As you can see, $ H^\* (\Omega G) = H^\* (\mathbb C \mathbb P^\infty ) $ for $\*\le 13$.
Even better: the varieties $\Omega G$ and $\mathbb C \mathbb P^\infty$ are isomorphic in complex dimensions $\le 6$ [*added later:* I take back that claim. I don't konw how to prove that the varieties $\Omega G$ and $\mathbb C \mathbb P^\infty$ are isomorphic in complex dimensions $\le 6$ (it might still be true)]. In particular, the *CW*-complexes $\Omega G$ and $\mathbb C \mathbb P^\infty$ are isomorhpic in dimensions $\le 13$.
Taking classifying spaces, we get that the *CW*-complexes $G$ and $K(\mathbb Z,3)$ are isomorphic in dimensions $\le 14$.


