Suppose we a given a fibration of manifolds $p\colon E\to M$ with a path connected fiber $F$ and simply connected $M$, then we have the Serre spectral sequence with

$$ E_2^{p,q} = H^p(M,\underline{H^q(F)}) $$

The standard proof of its convergence to $H^n(E)$ is purely topological and goes for singular or cellular cohomology. Can one give the proof in terms of de Rham cohomology?

In fact, I'm even more interested in de Rham cohomology with compact support. Now we have some difficulties in defining the local system as cohomology with compact support are no longer homotopy invariant but I hope these problems can be overcome.

EDIT. The proof for cohomology is given in Bott, Tu as V. Zaccaro told. In my answer I try to mimic the proof given there for cohomology with compact support but not completely successfully. I write the first sheet of the spectral sequence modulo two conjectures (I'd be very surprised if they're not true). I'd be grateful if you prove them and write the second sheet.

In the proof I use the notion of "quasi-sheaf with compact support" which I made up some days ago. Am I the first to introduce such a notion? If not, could you give me a reference on it?

R. Godement in "Topologie algébrique et théorie des faisceaux" uses another approach. He doesn't change the category ($Sh(X)$) but introduce another functor $\Gamma_{\Phi}$ where ${\Phi}$ is a family of supports, a family of compact sets in our case. Perhaps my proof can be retold in his language.

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    $\begingroup$ Differential Forms in Algebraic Topology-Bott and Tu : What you want is Theorem. 14.18 (Leray Theorem for de Rham Cohomology). The authors prove the theorem for fibers bundles, but they explain that it holds true for fibrations as well (p. 199). $\endgroup$ – Vincenzo Zaccaro Jan 6 '19 at 2:00

Let us first introduce the notion of a quasi-sheaf with compact support.

Def. Let $\mathcal F$ be a presheaf on a locally compact space $X$. We call it a quasi-sheaf with compact support if the following holds:

(A1) For every open $U\subset X$ and every section $s\in \mathcal F(U)$, $\overline{\mathrm{supp}(s)}$ is compact.

(A2) Let $U_i,i\in I$ be a family of opens, $s_i\in\mathcal F(U_i)$, for every two $i,j$ the restrictions of $s_i$ and $s_j$ to $U_i\cap U_j$ coincide and $\overline{\bigcup\limits_{i\in I}\mathrm{supp}(s_i)}$ is compact. Then there exist a section $s\in \mathcal F(\bigcup\limits_{i\in I}U_i)$ such that its restrictions on $U_i$ are $s_i$ and this section is unique.

Let us denote the category of quasi-sheaves with compact support on a locally compact $X$ by $QSh_c(X)$. Then the forgetting functor $QSh_c(X)\to PreSh$ has a left adjoint which we call a quasi-sheafication with compact support and denote by $\mathcal F^c$

Claim. $QSh_c(X)$ is an abelian category.

Let $p:X\to Y$ be a continuous map. Then it induces functors $p_*\colon Qsh_c(X)\to Qsh_c(Y)$ and $p^*\colon QSh_c(Y)\to QSh_c(X)$ if $p$ is proper..

Now we move to cohomology of qswcs.

Claim. $QSh_c(X)$ has enough injectives.

Def. (1) A qswcs is called flasque if $\forall U$ the map $\Gamma(\mathcal F)\to \mathcal F(U)$ is surjective.

(2) A qswcs is called soft if for every compact subset $K$ the natural map $\Gamma(\mathcal F)\to\Gamma(i_*i^*\mathcal F)$ where $i$ is the inclusion of $K$ in $X$ is surjective.

Claim. (1) Injective qswcs are flasque.

(2) Flasque qswcs on a paracompact space are soft.

(3) Flasque and soft qswcs on a paracompact space are acyclic.

Thm. Let $\underline{\mathbb R}$ denote the constant qswcs on a manifold $M$. Then $H^*(\underline{\mathbb R}) = H^*_c(M)$.

Now we are in position to compute the zeroth and first layer of Serre spectral sequnce. We can try to mimic the proof given in Bott, Tu for qswcs. The zeroth sheet of a spectral sequence is

$$E_0^{p,q} = \Gamma(\mathcal C^p(p_*\Omega^q(E)^c),$$

where $\mathcal C^q(\mathcal F)$ is the $q$-th term in the Čech resolvent of $\mathcal F$ associated with a cover of $M$ by small enough contractible compact subsets.

It converges to cohomology with compact support of $E$.

Now we compute the first sheet of the spectral sequence modulo two conjectures.

Firstly, we use the following conjecture which I can't prove formally but I think it follows from abstract homological algebra.

For each exact sequence of acyclic qswcs $\mathcal G^*$ we can form a qswcs in the following way. We have a family of presheaves $$ U \mapsto \frac{ker\: d\colon \mathcal G^n(U)\to \mathcal G^{n+1}(U)}{im\: d\colon \mathcal G^{n-1}(U)\to \mathcal G^n(U)} $$

Now take a quasi-sheafication with compact support of this presheaf and denote it by $\mathcal H^n(G^*)$.

Conj. (1) $H^n(\mathcal G^*):=\frac{ker\: d: \Gamma(\mathcal G^n)\to \Gamma(\mathcal G^{n+1})}{im\: d: \Gamma(\mathcal G^n)\to \Gamma(\mathcal G^{n+1})} = \Gamma(\mathcal H^n(\mathcal G^*))$

(2) $E_1^{p,q} = \Gamma(\mathcal C^p(\mathcal H^q(p_*\Omega^*(E))^c)$, i.e. taking cohomology commutes with forming the Čech complex.

Conj. Let $f,g\colon M\to N$ be two proper maps such that there exists a proper homotopy $H\colon M\times I\to N$ between them. Then they induce the same map on cohomology with compact support.

It follows that $\mathcal H^q(p_*\Omega^*(E)) = \underline{H_c^q(F)}$, hence

$$ E_1^{p,q}=\Gamma(\mathcal C^p(\underline{H_c^q(F)})^c) $$

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