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?

2 removed k-theory tag, added differential geometry tag
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# 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.