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

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?

• Not sure what "good/sensible (co)homology theory" is supposed to mean. Singular (co)homology ain't bad... – Kevin H. Lin Feb 1 '10 at 2:25
• Also have you seen this question? mathoverflow.net/questions/12920/… Some of the references there might be relevant. – Kevin H. Lin Feb 1 '10 at 2:36
• What properties do you want that singular (co)homology lacks? For most applications manifolds with corners are just manifolds with boundary with the boundary stratification modified slightly. – Ryan Budney Feb 1 '10 at 3:04
• Also, presumably you mean "(co)homology theory" in some unspecified but non-standard sense? The homotopy type of a manifold with corners is the same as its smoothing. Perhaps you want something multi-relative for the full inclusion of its stratifications, or something? – Ryan Budney Feb 1 '10 at 3:06
• I think what D. Thurston is getting at with his comment is that in order to show certain integral constructions are knot invariants you have to get some control of them at infinity, and when you compactify your space that amounts to understanding how the form behaves on the boundary strata. When working with explicit differential forms this is a lot of fussy work, but for example the point of view of Robin Koytcheff is another way to deal with this situation. – Ryan Budney Feb 1 '10 at 3:51

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)!

• Based on _coverDate=11%2F30%2F2002 in the URL, I conjecture that the broken link should point to: doi.org/10.1016/S0166-8641(02)00036-6 "Generalized billiard paths and Morse theory for manifolds with corners" David G.C.Handron, in Topology and its Applications, Volume 126, Issues 1–2, 30 November 2002, Pages 83-118. – j.c. Jul 17 '19 at 19:35

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

• (sorry if this comment comes many years later) - And how does the "diffeological" de Rham cohomology compare with the three options of @Orbicular's answer? – Qfwfq Nov 2 '18 at 9:11
• @Qfwfq I anticipate it should be the same as (1) and in particular give singular cohomology. – Mike Miller Nov 2 '18 at 14:57
• @ Qfwfq and @MikeMiller Actually we just published 2 papers on zero-perverse forms on stratified paces and diffeology. math.huji.ac.il/~piz/documents/DFOSS.pdf and math.huji.ac.il/~piz/documents/DFOSS-A.pdf We have also a paper on manifolds with corners and iffeology, not yet published math.huji.ac.il/~piz/documents/OMWBAC.pdf – Patrick I-Z – Patrick I-Z Nov 3 '18 at 14:40
• Thanks for the reference! I should have said "guess" instead of "anticipate" - clearly much work must be done. – Mike Miller Nov 3 '18 at 15:30