Why degree of a proper map is not a homotopy invariant on noncompact manifolds?

Question

I have read GTM218 Pg466, which argues that the degree of a proper map is not a homotopy invariant by showing the maps $F,G: \mathbb{C}\rightarrow\mathbb{C}$ given by F(z)=z and G(z)=z^2 are homotopy but have different degrees. But I'm wondering how does the compactness act in the proof of the degree of a map on the compact manifold is a homotopy invariant. I have a proof below: Since F is homotopy to G, then the induced map from $H^2_{dR}(\mathbb{C})$ to $H^2_{dR}(\mathbb{C}) F^{*}=G^{*}$. So for any 2-form w, $F^{*}w-G^{*}w=dv$, where $v$ is a 1-form. Let $k=degF$ and $t=degG$, we have $\int_{\mathbb{C}}F^{*}w=k\int_{\mathbb{C}}w$ and $\int_{\mathbb{C}}G^{*}w=t\int_{\mathbb{C}}w$. With Stokes theorem, we have $\int_{\mathbb{C}}(F^{*}-G^{*})w=\int_{\mathbb{C}}dv=\int_{\partial\mathbb{C}}v=0$ , I know this proof must be wrong, but what's the mistake?