Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $\mathfrak{g}$ be the Lie algebra of a Lie group $G$ which acts on a manifold $M$. It is quite standard that the basic forms in $\Omega^*(M) \otimes W(\mathfrak{g}^*)$ form a model for the singular equivariant cohomology of $M$. However, I have never seen a proof and it is not straightforward to me. Could someone give a sketch or a reference of the proof of this fact? It is probably in one of Cartan's papers but I haven't been able to find it.

Here goes some background:

We define its Weil algebra by $W^*(\mathfrak{g}^*)=S^*(\mathfrak{g}^*) \otimes \wedge^*(\mathfrak{g}^*)$ there is also a natural differential operator $d_W$ which makes $W*(\mathfrak{g}^*)$ into a complex. We define $d_W$ as follows:

Choose a basis $e_1,...,e_n$ for $\mathfrak{g}$ and let $e^*_1,...e^*_n$ its dual basis in $\mathfrak{g}^*$. Let $\theta_1,...,\theta_n$ be the image of $e^*_1,...e^*_n$ in $\wedge(\mathfrak{g}^*)$ and let $\Omega_1,...,\Omega_n$ be the image of $e^*_1,...e^*_n$ in $S(\mathfrak{g}^*)$. Let $c_{jk}^i$ be the structure constants of $\mathfrak{g}$, that is $[e_j,e_k]=\sum_{i=1}^nc_{jk}^ie_i$. Define $d_W$ by \begin{eqnarray} d_W\theta_i=\Omega_i- \frac{1}{2}\sum_{j,k} c_{jk}^i \theta_j \wedge \theta_k \end{eqnarray} and \begin{eqnarray} d_W\Omega_i=\sum_{j,k}c_{jk}^i\theta_j \Omega_k \end{eqnarray} and extending $d_W$ to $W(\mathfrak{g})$ as a derivation.

We can also define interior multiplication $i_X$ on $W(\mathfrak{g}^*)$ for any $X \in \mathfrak{g}$ by \begin{eqnarray} i_{e_r}(\theta_s)=\delta^r_s, i_{e_r}(\Omega_s)=0 \end{eqnarray} for all $r,s=1,...,n$ and extending by linearity and as a derivation.

Now consider $\Omega^*(M) \otimes W(\mathfrak{g}^*)$ as a complex. Using this definition of interior multiplication, together with the usual definition of interior multiplication on forms, we define the basic complex of $\Omega^*(M) \otimes W(\g^*)$:

We call $\alpha \in \Omega^*(M) \otimes W(\mathfrak{g}^*)$ a basic element if $i_X(\alpha)=0$ and $i_X(d \alpha)=0$. Basic elements in $\Omega^*(M) \otimes W(\mathfrak{g}^*)$ form a subcomplex which we denote by $\Omega^*_G (M)$.

The claim is that $H^*(\Omega^*_G (M))=H^*(M \times_G EG)$ where the right hand side denotes the singular equivariant cohomology of $M$.

share|improve this question
I cleaned up the TeX, and added in some \mathfrak's; the issue is that sometimes one has to put backticks around the TeX. –  Daniel Litt Oct 25 '11 at 0:54
add comment

1 Answer

up vote 2 down vote accepted

see the very nice book of Guillemin-Sternberg (Supersymmetry and ...); it also has a reprint of Cartan's paper.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.