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I have come heard that the cohomology of the classifying space of a compact torus $T$ is equal to the symmetric algebra over the dual of its Liealgebra $t^*$, where elements of the $t^*$ are of degree two:

$$H^*(BT)=S(t^*)$$

While I am able to compute that both sides are isomorphic to a polynomial ring in $dim(T)$ generators (of deg 2), I don't see at all how to write down a canonical map between both sides.

So my questions are:

What is the above map? How to think about it?

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up vote 4 down vote accepted

In your case, $BS^1$ is $\mathbb{CP}^\infty$. To see this, you can either use the fact that complex line bundles are classified by (homotopy classes of) maps into $\mathbb{CP}^\infty$ and that complex line bundles are the same as principal $S^1$-bundles (in one direction, choose a metric on your bundle and consider the sphere bundle), or you can use explicit cover $$S^{1} \to S^{\infty} \to \mathbb{CP}^\infty$$ (the infinite-dimensional Hopf map) to see that there is a $S^1$-bundle over projective space whose total space is contractible, which is the distinguishing property of the classifying space. The cohomology of $\mathbb{CP}^\infty$ can be computed explicitly (it's a polynomial ring on a generator in degree two).

If you work modulo torsion, i.e. with $\mathbb{R}$-coefficients, this map can be given by the Chern-Weil homomorphism (which has the advantage of generalizing to any $G$). If $G$ is any Lie group, then there is a morphism of algebras from the space of invariant polynomials on $\mathfrak{g}$ (i.e. $(\mathrm{Sym} \mathfrak{g}^*)^G$) to the cohomology ring $H^*(BG; \mathbb{R})$. Essentially, the idea is that cohomology classes of $BG$ are the same thing as "characteristic classes" of principal $G$-bundles (by the universal property of $BG$), so the point is to, for each space $M$ with a principal $G$-bundle $P \to M$, to assign a characteristic class in $H^*(M; \mathbb{R})$ measuring in some sense the nontrivialty of your bundle for each invariant polynomial $q: \mathfrak{g} \to \mathbb{R}$.

To construct a characteristic class on the manifold $M$ for the invariant polynomial $q$ and the bundle $P \to M$, one can work as follows: choose a connection on $P$. By (at least one) definition, this is a $\mathfrak{g}$-valued 2-form $\omega$ on $P$, with a certain equivariance property of the $G$-action. Then, consider $q(\omega)$ as an ordinary differential form on $M$. One can show that this descends to a form on $M$ (this comes from the invariance of the polynomial $q$), is closed, and that its cohomology class does not depend on the connection. (This is explained in Kobayashi-Nomizu's book on differential geometry.) This is the characteristic class you want, in de Rham cohomology. Note in particular that this gives you a way of computing Chern classes by "analytic" data (the first Chern class is the situation you asked about).

The Chern-Weil homomorphism is an isomorphism for a compact Lie group.

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Thanks very much Akhil. You mean the cohomology class of $q(\omega)$ does not depend on the connection right? –  jan Sep 24 '11 at 8:01
    
@Jan: Yes, that's right. –  Akhil Mathew Sep 24 '11 at 16:36
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