I'm looking for a book, article, or lecture notes that does basic cohomology theory from a relative point of view (including the Thom isomorphism, the excision theorem, Lefschetz duality, the Gysin sequence, etc.) and uses the de Rham model for relative cohomology.

Bott and Tu does most of basic cohomology theory using the de Rham model and even has a brief section on how to define the relative de Rham groups, but they mostly avoid the relative groups when formulating and proving the main results. Hatcher uses relative cohomology groups all over the place but doesn't really do anything with de Rham cohomology. I've been trying to build a dictionary between these two languages but I've run into some trouble at various points and I was hoping that somebody else has sorted all of this out.

  • $\begingroup$ De Rham groups (seem to) make no sense for arbitrary subsets of manifolds, so what kind of relative theory can one hope for? $\endgroup$ – Igor Belegradek May 4 '12 at 22:04
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    $\begingroup$ Bott and Tu define a relative cohomology group associated to a smooth map $f: M \to N$ between smooth manifolds: a relative cohomology class consists of a closed form on $N$ together with a reason why its pullback to $M$ is exact. So in particular one has a relative group $H^p(M,S)$ where $S$ is a smooth submanifold of $M$. $\endgroup$ – Paul Siegel May 5 '12 at 2:56
  • $\begingroup$ Have you looked at "From Calculus to Cohomology Theory" by Madsen and Tornehave? I don't have my copy in front of me, and a Google book search turns up only "relative compact" when searching for "relative," but the book does discuss the Thom isomorphism. Ranicki once told me that there was an on-line course in England (taught by Weiss, mainly, I recall) that took this perspective. Maybe there are still notes and/or lectures available on-line? I haven't a clue where to look, though. $\endgroup$ – Dan Ramras May 5 '12 at 4:03

That ain't easy. However, if you accept that DeRham cohomology is nothing but the cohomology of a particular soft resolution of the constant sheaf, then the book Cohomology of Sheaves by B. Iversen is of great help, albeit a bit hard to digest. The techniques in this book are really powerful and they've gotten me out of a jam many a times.

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    $\begingroup$ That's a very nice book. Maybe I should just try to learn that language instead... $\endgroup$ – Paul Siegel May 4 '12 at 21:00

If $N \subset M$ is a closed (meaning a closed subset, not a compact submanifold without boundary) submanifold, then the restriction map $\Omega^{\ast}(M) \to \Omega^{\ast}(N)$ is surjective. You can see this using local adapted charts and partitions of unity. Thus you can define the relative cohomology as the cohomology of the complex that is the kernel of the restriction map. This is, as far as I remember, the viewpoint in Jost "Riemannian Geometry and Geometric Analysis".


Manifold pairs is unfortunately a very limited context. Maybe one can also deal with images of immersions, and for most applications it should be enough. I personally prefer to go the other direction, to cohomology designed to deal with general spaces, such as sheaf cohomology or Alexander-Spanier cohomology. Liviu has commented on the former, so I discuss the latter.

The advantage of the Alexander-Spanier approach is that cohomology of the pair $(X,A)$ coincide with cohomology of the complement $X-A$. These are the so called "cohomology of one space" where the need of relative theory is largely eliminated. A standard treatment is in Massey's book "Homology and cohomology theory, an approach based on Alexander-Spanier cochains".


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