Normalized area integral equals the degree

If ${\phi }^{\mathrm{\prime }}=\phi +\delta \psi$ is a second representative of the same real cohomology class and $c^{\mathrm{\prime }}=c+\mathrm{\partial }b$ a second representative of the same real homology class, then ${\phi }^{\mathrm{\prime }}(c^{\mathrm{\prime }})=\phi (c)+\phi (\mathrm{\partial }b)+(\delta \psi )(c)+(\delta \psi )(\mathrm{\partial }b)=\phi (c)$: the three additional terms vanish respectively by $\delta \phi =0$, $\mathrm{\partial }c=0$, and ${\mathrm{\partial }}^2=0$. Thus the real evaluation in [F7], the real-bilinear version of the Kronecker pairing [F4], is independent of representatives. For every continuous $f:X\to Y$, the chain-level formulas $f^{\ast }\phi =\phi \circ f_{\mathrm{\#}}$ and $f_{\ast }c=f_{\mathrm{\#}}c$ give $(f^{\ast }\phi )(c)=\phi (f_{\mathrm{\#}}c)$, so $⟨f^{\ast }[\phi ],[c]{⟩}_{\mathbb{R}}=⟨[\phi ],f_{\ast }[c]{⟩}_{\mathbb{R}}$. Linearity of $\phi$ and $c$ makes the pairing real-bilinear.

In spherical coordinates $x=\sin \theta \cos \varphi$, $y=\sin \theta \sin \varphi$, $z=\cos \theta$, the unnormalized displayed two-form is $\sin \theta d\theta \wedge d\varphi$. Integrating for $0\le \theta \le \pi$ and $0\le \varphi \le 2\pi$ gives $4\pi$, so ${\omega }_{S^2}$ is a smooth closed form with integral one; its displayed ambient expression shows it extends smoothly across the coordinate poles.

Let $J$ be the natural comparison in [F3]. By [F5] and the real evaluation defined in [F7], ${\int }_{{\mathbb{T}}_{2\pi }^2}{n}^{\ast }{\omega }_{S^2}=⟨J([n^{\ast }{\omega }_{S^2}]),[{\mathbb{T}}_{2\pi }^2{]}_{\mathbb{R}}{⟩}_{\mathbb{R}}$. Naturality of $J$ and the real-chain pairing proved in step 1.1 gives ${\int }_{{\mathbb{T}}_{2\pi }^2}{n}^{\ast }{\omega }_{S^2}=⟨J([{\omega }_{S^2}]),n_{\ast }[{\mathbb{T}}_{2\pi }^2{]}_{\mathbb{R}}{⟩}_{\mathbb{R}}$.

By [F1], $n_{\ast }[{\mathbb{T}}_{2\pi }^2{]}_{\mathbb{Z}}=\mathrm{deg}(n)[S^2{]}_{\mathbb{Z}}$. The coefficient map $\iota :C_{\bullet }(X;\mathbb{Z})\to C_{\bullet }(X;\mathbb{R})$ sends each finite sum ${\sum }_j{a}_j{\sigma }_j$ to ${\sum }_j{a}_j{\sigma }_j$ with the same coefficients viewed in $\mathbb{R}$. It commutes with boundaries because [F6] defines the singular boundary as an integer linear combination of faces and $\iota$ is additive; it commutes with $n_{\mathrm{\#}}$ because [F6] postcomposes each simplex without changing its coefficient. Hence it induces a coefficient-change map on homology and carries each integral fundamental class to its real fundamental class: it sends the positive local orientation generator $1$ to $1$, so uniqueness of the class with those local restrictions in [F2] identifies the image. Applying $\iota$ to the degree equation gives $n_{\ast }[{\mathbb{T}}_{2\pi }^2{]}_{\mathbb{R}}=\mathrm{deg}(n)[S^2{]}_{\mathbb{R}}$. By the real bilinearity in [F7] and steps 2.1, ${\int }_{{\mathbb{T}}_{2\pi }^2}{n}^{\ast }{\omega }_{S^2}=\mathrm{deg}(n)⟨J([{\omega }_{S^2}]),[S^2{]}_{\mathbb{R}}{⟩}_{\mathbb{R}}=\mathrm{deg}(n){\int }_{S^2}{\omega }_{S^2}=\mathrm{deg}(n)$ by [F5] and step 1.2.