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I'm currently in need an explicit formula in classical cohomology which I'm pretty sure is well known, but which I've been unable to find in the references I am aware of.

Let $X$ be a smooth manifold and let $\mathcal{U}=\{U_\alpha\}$ be a fixed open cover of $X$ such that all the finite intersections $U_{\alpha_1}\cap\cdots U_{\alpha_n}$ are contractible. Consider the following two cochain complexes:

the de Rham complex $\Omega^\bullet(X)$ of $X$

the total complex of the Cech-de Rham bicomplex $\Omega^\bullet(\mathcal{U}_\bullet)$.

The restriction of a global form on $X$ to the open sets $U_\alpha$ gives a linear map

$j: \Omega^\bullet(X) \to Tot^\bullet(\Omega^\bullet(\mathcal{U}_\bullet))$

which, if I'm not wrong here, is a injective quasi-isomorphism of cochain complexes. I've been able to prove this (if I've not made mistakes), by brute force: i.e. by showing that $j$ is bijective in cohomology. But I'd like to have a fancier proof by writing an explicit "globalization" morphism

$\pi : Tot^\bullet(\Omega^\bullet(\mathcal{U}_\bullet)) \to \Omega^\bullet(X)$

such that

$\pi j= id_{\Omega^\bullet(X)}$

$j \pi = id_{Tot^\bullet(\Omega^\bullet(\mathcal{U}_\bullet))} + [d_{tot},K]$

with $K$ some explicit morphism of graded vector spaces $Tot^\bullet(\Omega^\bullet(\mathcal{U}_\bullet)) \to Tot^\bullet(\Omega^\bullet(\mathcal{U}_\bullet))[-1]$.

I guess one should be able to build $K$ by using a partition of the unit subordinate to the cover $\mathcal{U}$, but somehow I got lost in the computation. Since I feel this should be a well known fact, I'm asking here for direct references before attempting back to write $K$ myself.

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2 Answers

To check that $j$ is a quasiisomorphism it is enough to show that the cone of $j$, that is extended total Cech-de Rham complex, is acyclic. For this you can use the spectral sequence whish starts with the Cech differential. By Poincare Lemma its first page will be $0$, hence the spectral sequence converges to zero.

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Hi Sasha, thanks! That's a very elegant argument, which is actually a very neat and clear way of reorganize my brute force computation on cohomology classes. Thanks a lot! However what I'd really need here is an explicit homotopy $K$ (which I think one should be able to build by unwinding the proof that $j$ is a quasiisomorphisms, but which I also suspect is already written explicitly somewhere, that's why I'm asking for references here). –  domenico fiorenza Nov 27 '12 at 6:45
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up vote 4 down vote accepted

Thanks to an email by Chris Rogers, I now know that my question above is precisely the subject of Proposition 9.5 in Bott-Tu, Differential Forms in Algebraic Topology., where an explicit formula for the homotopy operator in terms of a partition of unit subordinate to the given open cover is given.

They also write "The not very intuitive formulas below were obtained, after repeated tries, by a careful bookkeeping of the inductive steps in the proof of Proposition 8.8." (where Proposition 8.8 is the statement that $j$ is a quasiisomorphism), which comforts me a lot in view of my failed attemps. Yet, since Bott-Tu is a basic reference on the subject I should have checked it, too, before asking on MO. Sorry for having not done.

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