In 1960 Borel proved a beautiful result:

Theorem. Let G be a simple, simply connected Lie group. Suppose that p is a prime that does not divide any of the coefficients of the highest root (expressed as a linear combination of the simple roots). Then $H^*G$ has no p-torsion.

Here $H^*G$ refers to the integral cohomology of $G$ as a space.

(Interestingly, the converse of this statement does not hold. For example $Sp(n)$ does not have $2$-torsion, though the coefficients of the highest weight are (1,2). This counterexample is given by Borel).

You can check out his proof here if you're interested, but basically it seems like what he does is this: he computes the cohomology of each of the simple, simply connected Lie groups with coefficients that have the relevant primes inverted and proves the result by observation.

Since the statement of the theorem itself has little to do with the actual classification of simple Lie groups, and morally should only rely on the fact that we can recover a simple, simply-connected Lie algebra from its root system, it seems natural to ask:

Question: In the intervening 50+ years since Borel proved this theorem, do we know of a direct proof of the theorem that does not use the classification of simple Lie groups?

I really have no idea how to go about doing this... could one use Lie algebra cohomology? I can only seem to find a link between Lie algebra cohomology and deRham cohomology of Lie groups, which is of no help when asking questions of torsion. Maybe there's an integral version?

1

# Proof for which primes H*G has torsion

In 1960 Borel proved a beautiful result:

Theorem. Let G be a simple, simply connected Lie group. Suppose that p is a prime that does not divide any of the coefficients of the highest root (expressed as a linear combination of the simple roots). Then $H^*G$ has no p-torsion.

Here $H^*G$ refers to the integral cohomology of $G$ as a space.

(Interestingly, the converse of this statement does not hold. For example $Sp(n)$ does not have $2$-torsion, though the coefficients of the highest weight are (1,2). This counterexample is given by Borel).

You can check out his proof here if you're interested, but basically it seems like what he does is this: he computes the cohomology of each of the simple, simply connected Lie groups with coefficients that have the relevant primes inverted and proves the result by observation.

Since the statement of the theorem itself has little to do with the actual classification of simple Lie groups, and morally should only rely on the fact that we can recover a simple, simply-connected Lie algebra from its root system, it seems natural to ask:

Question: In the intervening 50+ years since Borel proved this theorem, do we know of a direct proof of the theorem that does not use the classification of simple Lie groups?

I really have no idea how to go about doing this... could one use Lie algebra cohomology? I can only seem to find a link between Lie algebra cohomology and deRham cohomology of Lie groups, which is of no help when asking questions of torsion. Maybe there's an integral version?