In GTM82, I read a model for the relative cohomology of (M,N) with N a submanifold of M. 


And in the page: 
[Relative De Rham cohomologies](https://mathoverflow.net/questions/111059/relative-de-rham-cohomologies/111063#111063), 
I got to know that there is another model for relative cohomology by using differential forms of M, like those forms which restrict to 0 on N. 


Taladris called this Godbillon model. And Johannes Ebert said this model is used by jost in his book "Riemannian geometry and geometric analysis". But I can't find this model although I have checked that book(the 6th edtion) twice.


I wanna know in which paper,book, or article I can find the definition of the second model.



de rham model for relative cohomology
Suppose that $C$ is a closed subset of $M$. Denote by $\newcommand{\eO}{\mathscr{O}}$ $\eO$ its complement. 


The DeRham cohomology of $M$ is in fact the cohomology associated to a particular soft resolution of the constant sheaf $\newcommand{\ur}{\underline{\mathbb{R}}}$ $\ur$ on $M$.


To any sheaf $\newcommand{\eS}{\mathscr{S}}$ $\eS$ on $M$ we can associate a sheaf $\eS\_{\eO}$, also on $M$ whose stalk $\eS\_{\eO}(x)$ at $x\in M$ is $\eS(x)$ if $x\in \eO$, and $0$ if $x\in M\setminus \eO$. For any open subset $U\subset M$ the space $\Gamma(U, \eS\_{\eO})$ of sections of $\eS\_{\eO}$ over $U$ consists of the section $s\in \Gamma(\eO\cap U,\eS)$ such that the support of $s$ is **closed in $U$**. The operation $\eS\mapsto \eS\_{\eO}$ is an *exact functor*. Moreover if $\eS$ is soft, so is $\eS\_{\eO}$. Thus the DeRham resolution


$$ 0\to\ur\to\Omega^0\to\Omega^1\to\cdots, $$


$\Omega^k=$ the sheaf of smooth $k$-forms on $M$, produces a soft resolution of $\ur\_{\eO}$


$$ 0\to\ur\_{\eO}\to\Omega\_{\eO}^0\to\Omega\_{\eO}^1\to\cdots . $$


Hence, the cohomology of the sheaf $\ur\_{\eO}$ is computed by the the cohomology of the complex


$$ \Gamma(M, \Omega\_{\eO}^0)\stackrel{d}{\to}\Gamma(M, \Omega^1\_{\eO})\stackrel{d}{\to}\cdots, \tag{1} $$


where, as explained above $\Gamma(M, \Omega^k\_{\eO})$ consists of smooth $k$-forms on $\eO$ whose support is a closed subset in $M$. Equivalently $\Gamma(M, \Omega^k\_{\eO})$ consists of forms on $M$ whose supports do not intersect $C$. If $M$ is compact, then $\Gamma(M, \Omega^k\_{\eO})$ consists of form with compact support contained in $\eO$.


On the other hand, the cohomology of $\ur\_{\eO}$ can be identified with the relative cohomology $H^\bullet(M,C;\mathbb{R})$. The complex (1) is the a DeRham model for this cohomology. For more details I recommend the comprehensive book of Kashiwara and Schapira *Sheaves on Manifolds* or [my notes](http://www3.nd.edu/~lnicolae/Verdier-ams.pdf) which are less comprehensive, but may guide you through the literature. In particular, see Remark 2.17 of my notes.


