16-5. Suppose $M$ and $N$ are oriented, compact, connected, smooth manifolds, and $F,G:M\to N$ are homotopic diffeomorphisms. Show that $F$ and $G$ are either both orientation-preserving or both orientation-reversing. [Hint: use Theorem 6.29 and Stokes's theorem on $M\times I$.]
(Theorem 6.29 says any continuous homotopy can be upgraded to a smooth homotopy.)
A trivial solution to this problem would be to use the Poincare duality: if $M,N$ are manifolds of dimension $n$, then $H^n(M)= H^0(M)\cong\mathbb{R}\cong H^0(N)=H^n(N)$. By homotopy invariance, we get that $F^*=G^*$ on cohomology classes. By Stokes' theorem, we get that the integral of an exact form on a manifold without boundary is zero. Then $\int_MF^*\omega=\int_NG^*\omega$ for any volume form $\omega$ on $N$.
However, I want to do this problem according to the hint. Let $\omega$ be a volume form on $N$. Let $H:I\times M\to N$ be a smooth homotopy so that $H(0,-)=F$ and $H(1,-)=G$.
$$ \int_{M}G^*\omega - \int_{M}F^*\omega \overset{(\dagger)}{=} \int_{\partial(I\times M)} H^*\omega = \int_{I\times M} dH^*\omega = \int_{I\times M} H^*d\omega = \int_{I\times M} 0 = 0 $$ Thus, we recover the same result. The only equality that I'm a little sketched about is the first equality. Let $s$ be the coordinate on $I = [0,1]$. I'm sure it's true, because an "outward" pointing vector field at $\{1\}\times M$ would be $\partial/\partial s$ whereas an "outward" pointing vector at $\{0\}\times M$ would be $-\partial/\partial s$ so it seems obvious that I should "add" $\int_M G^*\omega$ and "subtract" $\int_M F^*\omega$.
But I'm slightly lost as to how one might show $(\dagger)$ formally. Should I be trying to write everything out in coordinates? Can someone give me a hint, or walk me through the details, on how to show the equality at $(\dagger)$ rigorously?
We covered orientations and integration in the last few weeks of class, and we haven't had the chance to do much practice with this topic, so I'm still a little uncomfortable doing these basic manipulations. Any help would be appreciated!