The kunneth formula gives that $H^k(X \times Y) = \bigoplus_{i+j = k} H^{i}(X) \otimes H^{j}(Y)$, where $X$ and $Y$ are both manifolds. I wonder whether this is true on the cochain level. More specifically we have $\pi_1$, $\pi_2$ projections onto $X$ and $Y$, is it true that every $k$ form on $X\times Y$ is of the form $\sum\pi_1^{\ast}(w_1) \land \pi_2^{\ast}(w_2)$?
Is the Kunneth formula for de Rham cohomology true on the cochain level
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Related Solutions
It is convenient for an answer to this question to generalise the usual homology $H_*(C,A)$ of a chain complex $C$ (of abelian groups, and with $C_n= 0$ for $n < 0$) from coefficients in an abelian group $A$, to the case where $A$ is also a similar chain complex. We work with the definition $H_*(C;A)=H_*(C \otimes A)$.
So we are thinking of the questioner's $H_n(B \times F;G)$ as $H_n(B ; C(F;G))$, where $C(F;G)$ is of course $C(F) \otimes G$, the chains of $F$ with coefficients in $G$.
Then we use three basic and well known properties of such chain complexes:
1) for any chain complexes $F,A $ such that $F$ is free, and morphism $\phi: H_*(F) \to H_*(A)$ of graded groups, there is a morphism $f: F \to A$ of chain complexes such that $H_*(f)=\phi$; in particular, if $F$ is free. there is a morphism $f: F \to H_*(F)$ such that $H_*(f)$ is the identity.
2) if $F$ is a free chain complex and $g: A \to B$ is a morphism of chain complexes such that $H_*(g): H_*(A) \to H_*(B) $ is an isomorphism, then $1\otimes g: F \otimes A \to F \otimes B$ induces an isomorphism of homology;
3) if $A$ is a chain complex there is a free chain complex $L$ such that there is a morphism $a: L \to A$ inducing an isomorphism in homology.
From this we deduce that if $A$ is a chain complex and $F$ is a free chain complex then there is an isomorphism $$\kappa_F:H_*(F;A) \to H_*(F;H_*(A))$$ which can be chosen to be natural with respect to maps of $F$. To get $\kappa_F$ we choose a free chain complex $L$ and a morphism $a: L \to A$ inducing an isomorphism in homology. Then we choose a morphism $b: L \to H_*(L) $ inducing an isomorphism in homology.
This can lead to specific calculations of $\kappa_F$.
This is the dual of arguments for cohomology in this paper Chains as coeficients, (Proc. LMS (3) 14 (1964) 545-65) and examples are given there of non naturality. The original problem as suggested by M.G. Barratt was to get some results on Postnikov invariants of function spaces $X^Y$ by induction on the Postnikov system of $X$.
The trick is that you can actually work globally instead of locally; all the local-to-global work can be done just on $M$ and $N$ themselves using their orientability. So, just let $\omega$ and $\eta$ be any nowhere vanishing top forms on $M$ and $N$ (these exist since $M$ and $N$ are orientable), so $\pi_1^*\omega\wedge \pi_2^*\eta$ is a nowhere vanishing top form on $M\times N$. Any other top form is then a scalar multiple of $\pi_1^*\omega\wedge \pi_2^*\eta$ at each point, and thus can be written in the form $h\pi_1^*\omega\wedge \pi_2^*\eta$ for some smooth function $h$.
(If you want $\omega$ and $\eta$ to be compactly supported as well, then you can choose them to have large enough supports so that $\pi_1^*\omega\wedge \pi_2^*\eta$ is still nonzero on the entire support of $\chi$.)
Best Answer
No, this is false. The easiest case to see it is $k=0$: then you would be claiming that every smooth function $f$ on $X\times Y$ can be written as a finite sum of functions $(g_i\circ\pi_1)\cdot (h_i\circ\pi_2)$ for smooth functions $g_i$ and $h_i$ on $X$ and $Y$ respectively. But this is false; for instance, writing $f_x(y)=f(x,y)$, this implies that the functions $f_x$ span a finite-dimensional vector space (since they are all linear combinations of the $h_i$), which is not true for all $f$ (exercise: find an explicit smooth function $f$ on $\mathbb{R}\times\mathbb{R}$ such that infinitely many of the functions $f_x$ are linearly independent).