I'm trying to work out Proposition (6.16) from Bott and Tu which states that $H^*_{cv}(M\times\mathbb{R}^n)$ is isomorphic to $H^{*-n}(M)$. The first group being forms compactly supported along the fibre.
The idea is to define chain maps from one space (of forms) to the other. One of the maps, $\pi_*$, is integration along the vertical direction and the other, $e_*$ is (presumably) given by wedging with $n$ many bump functions $e$ of integral $1$, one in each of the fibre variables.
It is quite clear that $\pi_*\circ{e_*}=id$. To show that the other composition is identity (on cohomology), I follow what the book does in earlier proofs and try to construct a homotopy operator $K:\Omega^*_{cv}(M\times\mathbb{R}^n)\rightarrow\Omega^{*-1}_{cv}(M\times\mathbb{R}^n)$ as follows.
$$K(\phi f(x,t)dt_I):=\sum_{i}^{}(\phi\int_{-\infty}^{t_i}f(x,t)dt_i(-1)^{i-1}dt_{I-{i}}-\phi\int_{-\infty}^{\infty}f(x,t)dt_i(-1)^{i-1}.\int_{-\infty}^{t_i}e(s)ds.dt_{I-i})$$ Where $\phi$ is a form pulled back from $M$, $I$ is a multi-index and $x$, $t$ denote dependencies on the horizontal and vertical directions respectively. (Note, this is indeed compactly supported along the fibre)
I've tried many times now to show $Kd-dK=1-e_*\circ{\pi_*}$ (up to a constant) without luck and I'm starting to think some special care needs to be taken in defining the maps. My calculations allow me to cancel almost everything in $dK-Kd$ to get $c(1-e_*\circ{\pi_*})$ and one other term. I can't get rid of that unwanted term and another problem is that the constant $c$ sometimes vanishes depending on how many terms there are in $I$ (When there are half as many as $n$, the dimension of the fibre).
I don't expect a complete computation ($dK$ and $Kd$ have 8 terms each) as an answer as that would be very tedious and messy (But it would be great if someone could do so!). But I hope someone who's done this before can tell me if this is indeed the approach. The books says this proposition 'carries over verbatim from (4.7)' which I find very upsetting.
To take this off of the un-answered queue - but (basically verbatim) as in the comments above.
Lemma Suppose $A,B$ and $C$ are differential objects (e.g., complexes), and pairs $$f:A\to B \text{ and } \phi:B\to A$$ and $$g:B\to C \text { and } \gamma:C→B$$
of maps commuting with $d$ - e.g., $df=fd$, where $d=d_B$ and $d=d_A$ as relevant, and so on, for $\phi$,$g$ and $\gamma$. Suppose there are homotopies $K:A→A$, and $L:B→B$: $$ Kd−dK=1−\phi f \text { and }Ld−dL= 1−\gamma g.$$
Then there is a homotopy $M:A\to A$: $$Md−dM=1−\phi\gamma gf,$$ where $M=\phi Lf+K.$
The proof of the lemma is immediate....
Using the lemma, one can collate homotopies, 'killing' one dimension at a time, i.e., by using the homotopy for one dimension of the book:
$$K_N:\Omega^*_{cv}(N\times\mathbb{R}^1)\rightarrow\Omega^{*-1}_{cv}(N\times\mathbb{R}^1),$$ with $N= M \times \mathbb R^k$, and $k = 0, \cdots,\ n-1$, where, in the notation of the original question, $K_N\phi f =0$, and (up to sign, depending on the weight of the form, if one wants to use the lemma above, exactly as stated), $$K_N \phi f(x,t)\,dt = \phi\left ( \int^{t}_{-\infty} f(x,s)\,ds - \int^t_{-\infty} e(s)\,ds \cdot \int^\infty_{-\infty} f(x,s)\,ds\right).$$