Let $M, N$ be compact manifolds and $\Omega^*$ its algebra exterior. How to prove that $\Omega^*(M)\otimes \Omega^*(N)$ is isomorphic to $\Omega^*(M\times N)$?

I thought about the function $f(\omega,\eta) = \pi_M^*(\omega)\wedge \pi_N^*(\eta)$, where $\omega \in \Omega^*(M)$, $\eta \in \Omega^*(N)$ and $\pi_M$, $\pi_N$ the canonical projection.

The problem is to construct the linear function define on the tensor product. How to proceed?

  • $\begingroup$ Can you think in a bilinear map from $\Omega(X)\times\Omega(Y) \to \Omega(X \times Y)$? If yes, then you should use the universal property of the tensor product to finish... think you can proceed from here? EDIT: Tipo $\endgroup$
    – User43029
    Jun 13, 2016 at 22:48

1 Answer 1


Such an isomorphism is not true. What really happens is that, under the bilinear map from $\Omega^*(M)\times\Omega^*(N)$ into $\Omega^*(M\times N)$ you wrote (which, by universality of the tensor product, lifts uniquely to a linear map from $\Omega^*(M)\otimes\Omega^*(N)$ into $\Omega^*(M\times N)$ as user 123456 suggested), $\Omega^*(M)\otimes\Omega^*(N)$ is isomorphic to a dense subspace of $\Omega^*(M\times N)$ in the (Fréchet) topology of uniform convergence of all derivatives in $M\times N$ (or compact subsets thereof if $M$ or $N$ is not compact; it is not difficult to see in either case that this topology does not depend on the choice of atlas).

The trouble begins already at forms of degree zero: it is not true that every $F\in\mathscr{C}^\infty(M\times N)$ is of the form $$F(x,y)=\sum^n_{j=1}f_j(x)g_j(y)\ ,\quad f_j\in\mathscr{C}^\infty(M)\ ,\,g_j\in\mathscr{C}^\infty(N)$$ for some $n\in\mathbb{N}$. Take for instance $M=N=\mathbb{R}/2\pi\mathbb{Z}$ and $$F(x,y)=\sum^\infty_{m,n=0}a_{mn}\cos(mx)\cos(ny)\ ,$$ where the double sequence $(a_{mn})_{m,n\geq 0}$ decreases to zero faster than any negative power of $m+n$ but never becomes zero as $m,n\to+\infty$, guaranteeing that $F$ is smooth and doubly periodic - e.g. $a_{mn}=e^{-mn}$ (the same happens if $M$ or $N$ is not compact, take e.g. $M=N=\mathbb{R}$ and $F(x,y)=e^{xy}$).

(Note: it is also not difficult to check that the linear map obtained from the bilinear map you wrote is injective. Just work pointwise using linear frames in the cotangent space)


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .