Let $G$ be a compact Lie group and $\pi:P \to M$ a principal $G$-bundle. I would like to understand the geometry of $M$ through $P$ with the $G$-action.

I am trying to understand the Hopf bundle ($G=U(1)$, $P=S^3$ and $M=S^2$) as an example, however I keep general notion in this question.

The de Rham complex $(\Omega^*(M),d)$ of $M$ is isomorphic to the complex $(\Omega^*(P)_b,d)$ of basic forms on $P$. Therefore the Euler number of the latter $$\sum_i(-1)^i\dim H^i(\Omega^*(P)_b,d) \tag{E}$$ is equal to one of the former: $\chi(M)$.

Is it possible to compute (E) without comparing the de Rham complex of $M$?

$\Omega^k(P)_b$ is the space of $G$-invariant sections of $\wedge^k H$, where $H$ is the vector bundle of horizontal forms on $P$. One of the difficulties is that the space $\Gamma(P,\wedge H)$ is not closed under $d$, while $\Gamma(P,\wedge H)^G$ ($=\Omega^*(P)_b$) is closed under $d$. Therefore we can not apply $G$-equivariant index theorem (after $\otimes \mathbb C$).