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Let $V$ be a $n$-dimensional vector space, acted on by a compact Lie group $G$. We denote by $V_G := V \times EG / G$ the homotopy quotient, where $EG \to BG$ is the universal principal bundle associated with $G$.

Are there canonical representatives of the Thom and Euler classes of the vector bundle $V_G \to BG$ ?

So far, I have found a construction of Mathaï and Quillen of a universal equivariant Thom form in the context of equivariant De Rham cohomology, using the Weil and Cartan models of equivariant differential forms.

On the other hand, It I think (please correct me if I'm wrong) that it should be possible, on any vector bundle $E \to B$ over a nice enough base, to construct a Thom form by means of a global angular form. On $V_G \to BG$, we might be able to use the same construction, filtrating $EG$ and $BG$ by finite dimensional spaces.

I have trouble putting some order in all these constructions, and would be very grateful if someone could clarify the relations between them.

Thanks a lot !

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