Hello,

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 $i:M\to N$ 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.