Suppose you have a closed manifold $M$ and a function $f:M\times M\rightarrow\mathbb{C}$ that is $C^2$ in the $M$ variables, and suppose that at any point $q\in M$, we have $$f(\cdot,q)\in H_m(M).$$ Is it possible to show that $f\in H_m(M\times M)$?
I feel like there should be an easy way to show this using compactness and continuity in the second factor, but the precise expression eludes me.
Cheers.