I am reading Jean-Luc Brylinski's book on loop space. At the end of section 1.6 Leray Spectral Sequence, he claims without proof that
Let $f: Y\to X$ be a smooth bundle of paracompact manifolds. A $p$-form $\omega$ on $Y$ is called basic if it satisfies:
For any vertical vector field $\xi$ on $Y$,
- $i(\xi)\omega =0$;
- $i(\xi)\mathrm{d} \omega =0.$
Assume the fiber $F$ is connected, then $f^*$ identifies space of smooth $p$-forms $A^p(X)$ with space of basic $p$-forms on $Y$.
I wonder whether it can be proved by Cartan homotopy formula.
More precisely, given $\omega$ basic, define $\eta (x_0) =\omega (x_0, z)$ for $(x_0,z)\in f^{-1}(x_0)$. Choose a local coordinate we see that $\omega$ has no fiber components.
Since $L_{\gamma '(t)}\omega =0$ for path in $f^{-1}(x_0)$, it is well-defined.
Does this work? The following are some info that I thought might help.
A somewhat related post is here.
Brylinski also describes basic forms in language of Cartan filtration $F^k$, which is subcomplexes of $A^{\bullet}(Y)$ where $F^k(A^n(Y))$ consists of those $n$-forms $\omega$ such that for any vertical vector fields $\xi _1, \cdots ,\xi _{n-k+1}$, $$i(\xi _1)\cdots i(\xi _{n-k+1})\omega=0.$$
I looked up some related books. Brylinski referred Cartan filtration to Cartan's article La transgression dans un groupe de Lie et dans un fibré principal .
The same filtration appears in Akio Hattori's paper Spectral Sequence in the de Rham Cohomology of fibre bundles, in which Hattori referred that filtration to
- Leray L'anneau spectral et l'anneau filtré d'homologie d'un espace localement compact et d'une application continue
- Serre Homologie singulière Des espace fibrés
But I can hardly read French.