2 added 2 characters in body

Let $G$ be a compact Lie group, let $T$ be a maximal torus, and let $W$ be the Weyl group. My main question is as follows:

• How does one prove that $H^\ast(BG,\mathbb{Q})$ is isomorphic to the $W$-invariant part of $H^\ast(BT,\mathbb{Q}) \cong \mathbb{Q}[[x_1, \ldots, x_n]]$? This is apparently basic knowledge in algebraic topology, because I keep reading "recall that..." followed by some version of this statement and no references. But I can't find a proof in any of my textbooks.

I would ideally like a reference which also addresses the following secondary question:

• When is the natural map $H^\ast(BG,\mathbb{Z}) \to H^\ast(BT,\mathbb{Z})^W$ an isomorphism, and what can one say about the integral cohomology ring of $BG$ when it is not? Note the fact that the map above is an isomorphism for $G = U(n)$ is equivalent to the statement that the Chern classes are integral.

Thanks!

1

# Cohomology ring of BG

Let $G$ be a compact Lie group, let $T$ be a maximal torus, and let $W$ be the Weyl group. My main question is as follows:

• How does one prove that $H^\ast(BG,\mathbb{Q})$ is isomorphic to the $W$-invariant part of $H^\ast(BT,\mathbb{Q}) \cong \mathbb{Q}[[x_1, \ldots, x_n]]$? This is apparently basic knowledge in algebraic topology, because I keep reading "recall that..." followed by some version of this statement and no references. But I can't find a proof in any of my textbooks.

I would ideally like a reference which also addresses the following secondary question:

• When is the natural map $H^\ast(BG,\mathbb{Z}) \to H^\ast(BT,\mathbb{Z})^W$ an isomorphism, and what can one say about the integral cohomology ring of $BG$ when it is not? Note the fact that the map above is an isomorphism for $G = U(n)$ is equivalent to the statement that the Chern classes are integral.

Thanks!