TITLE: Relative De Rham cohomologies
QUESTION [33 upvotes]: as far as I know, there are two main ways to have a relative version of De Rham Cohomology for a pair (M,N), where M and N are smooth manifolds and N is a closed (as a topological subspace) submanifold of M:
1) Godbillon, Elements de topologie algébrique: $\Omega^p(M,N)$ is the space of all forms on $M$ whose restriction to $N$ is zero. This is a subalgebra of $\Omega^p(M)$ so it defines a cohomological space $H^p(M,N)$.
2) Bott-Tu, Differential forms in algebraic topology: this times, $\Omega^p(M,N)=\Omega^p(M)\oplus  \Omega^{p-1}(N)$ with differential $d(\omega,\theta)=(d\omega,i^*(\omega)-d\theta)$, where $i:N\to M$ is the inclusion.
Does these two cohomologies give the same results? Otherwise, are they related and how are they related?
Bott and Tu's paragraph on the relative De Rham cohomology is very short. Does someone know a good reference on this subject? 
Thank you in advance.

REPLY [4 votes]: AG learner's is almost the answer I wanted, but really bugged me that that $\Psi$ wasn't a chain map that was homotopic to the identity.  I think I figured out how to make that happen.  Let $T$ be an open tubular neighborhood of $N$, and $r\colon R\to N$ a smooth deformation retract.     Let $f$ be a smooth bump function whose support is in $T$ and which is 1 on $N$.  We can find the formula for $\Psi$ by considering the homotopy $h(\alpha, \beta)=(fr^*\beta,0)$.  We have that $$(dh+hd)(\alpha,\beta)=(d(fr^*\beta)+fr^*(\alpha|_N-d\beta),\beta)$$ so
$$(1-dh-hd)(\alpha,\beta)=(\alpha-fr^*(\alpha|_N)-df\wedge r^*\beta,0).$$
This coincides with AG learner's formula on closed forms, and is a chain map.