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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 $\Omega^*(M) \otimes W(\mathfrak g^*)$:

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 $M$.

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

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As mentioned by the user SGP, the book Supersymmetry and Equivariant de Rham Theory by Victor W Guillemin and Shlomo Sternberg discuss about Cartan model. One of the intentions is to prepare the reader to understand Cartan's papers:

  • Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie, Colloque de Topologie, C.B.R.M., Bruxelles 15-27 (1950)
  • La transgression dans un groupe de Lie et dans un espace fibré principal, Colloque de Topologie, C.B.R.M., BruxeIles 57-71 (1950)

Preface of the book :

This is the second volume of the Springer collection Mathematics Past and Present. In the first volume, we republished Hörmander's fundamental papers Fourier integral operators together with a brief introduction written from the perspective of 1991. The composition of the second volume is somewhat different: the two papers of Cartan which are reproduced here have a total length of less than thirty pages, and the 220 page introduction which precedes them is intended not only as a commentary on these papers but as a textbook of its own, on a fascinating area of mathematics in which a lot of exciting innovation have occurred in the last few years. Thus, in this second volume the roles of the reprinted text and its commentary are reversed. The seminal ideas outlined in Cartan's two papers are taken as the point of departure for a full modern treatment of equivariant de Rham theory which does not yet exist in the literature.

Introduction :

The year 2000 will be the fiftieth anniversary of the publication of Henri Cartan's two fundamental papers on equivariant De Rham theory "Notions d'algèbre différentielle; applications aux groupes de Lie et aux variétés où opère un groupe de Lie" and "La trangression dans un groupe de Lie et dans un espace fibré principal." The aim of this monograph is to give an updated account of the material contained in these papers and to describe a few of the more exciting developments that have occUfred in this area in the five decades since their appearance.

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