Help to correct answer: Product of orientable manifolds is orientable

Question

In the first reply to this post, I made a comment, but nobody answered me yet. Link: Orientability of a product of smooth manifolds implies orientability of each factor

My problem is in the return of the affirmation $(\Leftarrow)$.

Problem statement:

Given two smooth manifolds $M$ and $N$, show that the product manifold $M \times N$ is orientable if and only if $M$ and $N$ are orientable.

First reply from rmdmc89:

$1)$ If $M^m,N^n$ are both orientable, there are volume forms (i.e., non-zero top-forms) $\omega\in\Omega^m(M)$ and $\sigma\in\Omega^n(N)$. Define $\eta\in\Omega^{n+m}(M)$ as: $$\eta(X_1,...,X_n,Y_1,...,Y_m):=\omega(X_1,...,X_n)\sigma(Y_1,...,Y_m)$$

which is a volume form on $M\times N$, so $M\times N$ is orientable.

$2)$ Conversely, if $M\times N$ is orientable, there is a volume form $\eta\in\Omega^{n+m}(M\times N)$. Fix a point $q\in N$ and a basis $\left\{\left.\frac{\partial}{\partial y_1}\right|_q, ...,\left.\frac{\partial}{\partial y_n}\right|_q\right\}$ a for $T_qN$. Define $\omega\in\Omega^m(M)$ as: $$w(X_1,...,X_n):=\eta_{(\cdot,q)}\left(X_1,...,X_n,\left.\frac{\partial}{\partial y_1}\right|_q,...,\left.\frac{\partial}{\partial y_m}\right|_q\right)$$

which is a non-zero top-form on $M$, so $M$ is orientable. We can make a similar construction for $N$.

My comment on the first answer:

"I think it's wrong at the beginning of "Define $η∈Ω^{n+m}(M)$" must be "Define $η∈Ω^{n+m}(M\times N)$". And even then, I think the product is not well defined. In $Λ^{m+n}(M×N)$ has more vectors than ωσ can actually receive. Am I right? Think of the case $m=n=1$ you will see. I have a problem similar to posted, my idea was to do as you but I found this problem. Am I correct?"

How to correct?

Answer 1

You’re right that $\eta \in \Omega^{n+m} (M \times N)$. The product is fine as defined . Maybe you’d like a different formulation. Let $\omega \in \Omega^m(M)$ and $\sigma \in \Omega^n(N)$ be two volume forms. Let $\pi_M : M\times N \to M$ and $\pi_N : M\times N \to N$ be the canonical projection maps. Then let $\eta = \pi_M^\ast(\omega) \wedge \pi_N^\ast(\sigma)$. This is a volume form on $M \times N$ and is more or less the $\eta$ defined in the original response.

Edit: As Shifrin points out below, the form $\eta$ in the original answer doesn’t make sense. I think my construction of a volume form is correct however.