I recently ran across a qual question about a generalization the Hopf invariant to smooth maps $f: M^{4n-1} \to N^{2n}$ between arbitrary closed connected oriented manifolds of the indicated dimensions: Choose an $2n$-form $\beta$ on $N$ such that $\int_N \beta=1$. If $f^*\beta$ is exact, i.e. $f^* \beta = d \alpha$, then we can define $$H(f) = \int_M \alpha \wedge f^* \beta.$$
In particular, this generalizes Whitehead's integral expression for the Hopf invariant seen here.
Is this a widely-known definition? A quick literature search fell flat, so I would appreciate any pointers. If they are helpful, here are more specific questions: Is it $\mathbb{Z}$-valued? Does it have any useful applications (when $f$ is not a map of spheres)? Is there a corresponding generalization in the language of cohomology? Is it ever actually computed?
Update: In fact, I would be happy if anybody has an example of a smooth map of appropriate manifolds $f: M^{4n-1} \to N^{2n}$ and a $2n$-form $\beta$ on $N$ such that $f^*\beta$ is exact (and nontrivial).