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,
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.
Best Answer
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 which are less comprehensive, but may guide you through the literature. In particular, see Remark 2.17 of my notes.