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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 $Lie(S^1\ltimes$Lie(S^1\ltimes \widetilde {LG})^* \twoheadrightarrow (\mathfrak t_{S^1\ltimes \widetilde {LG}})^* \cong \mathfrak t^* \oplus \mathbb R \oplus \mathbb R$. 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. 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. 2 added 3049 characters in body I'll elaborate once 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.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 Rcalled 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). Here we go: 0-dimensional cell:0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 0 - 0 - 0 - 0\end{matrix} 2-dimensional cell:0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 0 - 0 - 0 - 1\end{matrix} 4-dimensional cell:0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 0 - 0 - 1 - 1\end{matrix} 6-dimensional cell:0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 0 - 1 - 1 - 1\end{matrix} 8-dimensional cell:0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 1 - 1 - 1 - 1\end{matrix} 10-dimensional cell:0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 1 - 1 - 1 - 1 - 1\end{matrix} 12-dimensional cell:0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{1} - 1 - 1 - 1 - 1 - 1\end{matrix} 14-dimensional cell:0 - 0 - \stackrel{\stackrel{\displaystyle 1}|}{1} - 1 - 1 - 1 - 1 - 1\end{matrix} other 14-dimensional cell: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. In particular, the CW-complexes \Omega G and \mathbb C \mathbb P^\infty are isomorhpic in dimensions \le 13.Taking classifying spaces, we get some more time..that the CW-complexes G and K(\mathbb Z,3) are isomorphic in dimensions \le 14. 1 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\$.

I'll elaborate once I get some more time...