Smooth homotopic maps and closed forms..

Question

Does anyone have any idea for showing the following: Let $f_0, f_1:M\rightarrow N$ smooth homotopic maps between the manifolds $M$ and $N$. Suppose $M$ is compact with no boundary. Show that for every closed form $\omega\in \Omega^m(N)$ (where $m=\dim M$), $$\int_{M}f_0^*\omega=\int_Mf_1^*\omega.$$

Answer 1

Let $F:M\times I\rightarrow N$ be a smooth homotopy between $f_0$ and $f_1$.

Then, since $d$ commutes with pullbacks and by Stokes's theorem, we have

\begin{align*} 0 &= \int_{M\times I} F^\ast d\omega \\ &= \int_{M\times I} dF^\ast \omega\\ &= \int_{\partial(M\times I)} F^\ast \omega \\ &= \int_{M\times\{1\}} F^\ast \omega - \int_{M\times\{0\}}F^\ast \omega\\ &= \int_M f_1^\ast\omega - \int_Mf_0^\ast \omega. \end{align*}