0
$\begingroup$

Let say I want to show two differential forms $\omega_1$ and $\omega_2$ on a smooth manifold $M$ are equal. Of course it suffices to show $\omega_1=\omega_2$ locally, i.e. the equality holds over every open subset $U_\alpha$ of a smooth atlas $\{(U_\alpha, \varphi_\alpha)\}$ of $M$.

Since each $U_\alpha$ is a smooth manifold given by an atlas consisting of a single chart, $U_\alpha$ is orientable (however, $M$ is NOT assumed to be orientable, or the charts of the atlas $\{(U_\alpha, \varphi_\alpha)\}$ are NOT assumed to be consistently oriented). Now suppose I have shown that $\omega_1=\omega_2$ on each $U_\alpha$, where the proof involves Poincare duality: $H^k(U_\alpha)\cong H_c^{n-k}(U_\alpha)^*$, where $n=\dim(M)$ and $0\leq k\leq n$. I am not sure if I can conclude that $\omega_1=\omega_2$ on $M$.

I have seen some arguments of proving two differential forms are equal, which is roughly the same as above: prove the equality locally, and locally one assumes the orientability, and it is used in an essential way (and with some other topological condition rather than orientability).

The reason I am not sure whether my argument in above is correct or not is, well, I feel like it's cheating, in the sense that Poincare duality is kind of topological. Of course, Poincare duality holds for non-orientable manifolds (whose statement involves the orientation bundle of $M$), but I cannot use this version of Poincare duality in my argument for some reason.

$\endgroup$
2
  • 1
    $\begingroup$ If you've shown two forms are equal on charts then they're equal, no matter what you used. On the other hand it's hard to imagine using Poincare duality here since that most likely only shows equality mod exact forms. (Unless your forms satisfy some other, unstated, condition like being harmonic etc) $\endgroup$ May 17, 2023 at 14:39
  • $\begingroup$ @OtisChodosh Thanks for your answer. The whole story is much more involved, and you are right that the forms do satisfy some other unstated conditions. $\endgroup$
    – Ho Man-Ho
    May 17, 2023 at 15:08

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.