To study the topology of Lie groups, you can decompose them into the simple compact ones, plus some additional steps, such as taking the cover if necessary. After that, the structure of $SO(n)$'s is rather straightforward, but the exceptional groups are more interesting.
Any simple compact Lie group, by means of Hopf algebra theory, has the rational homology of a product $$S^a \times S^b \times \dots \times S^z$$ where the numbers are called exponents. Other than that, their cohomology could also have torsion. Now the torsion for all groups is known:
- Among classical groups, only 2-torsion is possible and only for $Spin(n)$
- Exceptional groups can only have 2 and 3-torsion (most do), with the exception of:
- $E_8$ which has 2-, 3-, and 5- torsion.
Well, this is bound to be related to $E_8$'s Coxeter number, which is 30, but are there any hints as to why? My reference would be math-ph/0212067 but it can't relate this to Coxeter number either.
For the reference, exponents are known to be related to Coxeter number, see Kostant, The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group (google search).
Is this an open problem? Maybe yes, but maybe it's been explained, so I'm posting it as it is for now.