Is there a good (co)homology theory for manifolds with corners?

Question

Recall that a (smooth) manifold with corners is a Hausdroff space that can be covered by open sets homeomorphic to $\mathbb R^{n-m} \times \mathbb R_{\geq 0}^m$ for some (fixed) $n$ (but $m$ can vary), and such that all transition maps extend to smooth maps on open neighborhoods of $\mathbb R^n$.

I feel like I know what a "differential form" on a manifold with corners should be. Namely, near a corner $\mathbb R^{n-m} \times \mathbb R_{\geq 0}^m$, a differential form should extend to some open neighborhood $\mathbb R^{n-m} \times \mathbb R_{> -\epsilon}^m$. So we can set up the usual words like "closed" and "exact", but then Stokes' theorem is a little weird: for example, the integral of an exact $n$-form over the whole manifold need not vanish.

In any case, I read in D. Thurston, "Integral Expressions for the Vassiliev Knot Invariants", 1999, that "there is not yet any sensible homology theory with general manifolds with corners". So, what are all the ways naive attempts go wrong, and have they been fixed in the last decade?

As always, please retag as you see fit.

Edit: It's been pointed out in the comments that (1) I'm not really asking about general (co)homology, as much as about the theory of De Rham differential forms on manifolds with corners, and (2) there is already a question about that. Really I was just reading the D. Thurston paper, and was surprised by his comment, and thought I'd ask about it. But, anyway, since there is another question, I'm closing this one as exact duplicate. I'll re-open if you feel like you have a good answer, though. -Theo Edit 2: Or rather, apparently OP can't just unilaterally close their own question?

Answer 1

I suppose you are talking about deRham cohomology. Then it would be wise to take a look at the work of Richard Melrose, e.g. his book The Atiyah-Patodi-Singer Index Theorem.

On page 65 he discusses deRham cohomology for manifolds with boundary (which can be easily generalized to the corner case, as was also done by him!). On manifolds with corners something interesting happens: there are different versions of reasonable vector fields (and—by duality—differential forms), e.g.

1. extendible vector fields (like you mentioned)
2. tangent vector fields (tangent to any boundary hypersurface)
3. "zero" vector fields (vanishing on all boundary hypersurfaces)

(It can be shown that $d$ preserves the classes 1.-3., giving a deRham complex whose cohomology can be computed) Melrose points out (compact with boundary case) that in cases 1. and 3. the deRham cohomology is canonically isomorphic to the singular cohomology of the underlying topological space. For the 2nd case also the cohomology of the boundary enters (via a degree shift).

I should also point out that there is also a working Morse theory on manifolds with corners, see for example

M. Shida, Fundamental Theorems of Morse Theory for Optimization on Manifolds with Corners, Journal of Optimization Theory and Applications 106 (2000) pp 683-688, doi:10.1023/A:1004669815654

and [this broken link EDIT perhaps someone else can extract a result -DR]

Furthermore it is easy to construct "invariants" of the manifold with corners by also taking into account its corners (but be careful with respect to which transformations this is an invariant)!

Answer 2

A manifold with corner is a diffeological space modeled on orthants, as such it has a very well defined De Rham cohomology.

Edit : With Serap Gürer, we just wrote a paper (will appear in Indag. Math.) about differential forms on manifolds with corners. Here: http://math.huji.ac.il/~piz/documents/DFOMWBAC.pdf