Relative De Rham cohomologies

Question

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.

Answer 1

A chain map $\Theta$ from the Godbillon theory to the Bott-Tu version is given by $\omega \mapsto (\omega,0)$ (note that is a chain map only on $\Omega^{p} (M;N)_{G}$). I claim that this induces an isomorphism on cohomology. A couple of special cases is obvious: if $N=\emptyset$, then both theories agree with absolute de Rham theory. If $N \to M$ is a homotopy equivalence, both theories are trivial by long exact sequences and homotopy invariance of the absolute theory.

For the general case, pick a tubular neighborhood $U$ of $N$. You get short exact sequences of chain complexes (in both cases)

$$ 0\to \Omega (M;N) \to \Omega(U;N) \oplus \Omega (M-N) \to \Omega (U-N) \to 0 $$

(exactness is checked by means of a partition of unity), and $\Theta$ compares the both short exact sequences. The associated (Mayer-Vietoris) exact sequence and the $5$-lemma concludes the proof.

Answer 2

The fact that the DeRham cohomology of $M$ computes the singular cohomology is a reflection of the fact that the cohomology of a sheaf (in this case the constant sheaf $\newcommand{\bR}{\mathbb{R}}$ $\underline{\bR}$ over $M$) can be computed by using any (soft, flabby) resolution of this sheaf. The geometry appears when we produce a concrete such resolution, in your case the DeRham resolution.

Thus, an appropriate question would be if the relative cohomology could be given a sheaffy description.

The answer is yes. If $\newcommand{\eS}{\mathscr{S}}$ $\eS$ is a sheaf of Abelian groups on $M$ and $Z\subset M$ is a closed subset, then we denote by $\eS_Z$ the ``restriction'' of $\eS$ to $Z$ extended by $0$ outside. I want to emphasize that $\eS_Z$ is a sheaf on $M$: its stalk at $x\in M$ is $\eS_x$ if $x\in Z$, and $0$ otherwise. We have a natural surjective morphism

$$ r:\eS\to\eS_Z $$

We denote its kernel by $\eS_{M\setminus Z}$.We thus get a short exact sequence of sheaves

$$ 0\to\eS_{M\setminus Z} \to \eS\to\eS_Z\to 0. $$

This leads to an exact sequence in cohomology

$$ \cdots \to H^\bullet(M,\eS_{M\setminus Z})\to H^\bullet(M,\eS)\to H^\bullet(M,\eS_Z) \stackrel{+1}{\to}\cdots $$

When $\eS=\underline{\bR}$ is the constant sheaf on $M$ with stalk $\bR$ then $H^\bullet(M,\underline{\bR})$ is the singular cohomology of $M$, $H^\bullet(M,\underline{\bR}_Z)$ is the cohomology of $Z$ and $H^\bullet(M,\underline{\bR}_{M\setminus Z})$ is the (relative) cohomology of the pair $(M,Z)$.

You can get the two descriptions by from this fact alone (the one in Bott-Tu requires additionally the cone of construction). I refer to Iversen's book Cohomology of sheaves.