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In $\mathbb{S}^1$ we have the standard $2\pi$-periodic parameterization $\mathbb{R}\longrightarrow\mathbb{S}^1$, $\theta\longmapsto \mathrm{e}^{\mathrm{i}\theta}$. This parameterization induces a global 1-form, $\mathrm{d} \theta$, and $C^{\infty}(\mathbb{S}^1)$ is identified with the set of differentiable and $2\pi$-periodic functions $\mathbb{R}\longrightarrow\mathbb{R}$

  • 0-forms: $f:\mathbb{R}\longrightarrow\mathbb{R}$ differentiable and $2\pi$-periodic
  • 1-forms: $g\mathrm{d} \theta$, $g:\mathbb{R}\longrightarrow\mathbb{R}$ differentiable and $2\pi$-periodic
  • $\mathrm{d}(f)=\mathrm{d} f=\dfrac{\partial f}{\partial \theta}\mathrm{d} \theta$
  • $\mathrm{d}(g\mathrm{d}\theta)=\mathrm{d} g\wedge\mathrm{d} \theta=\dfrac{\partial g}{\partial \theta}\mathrm{d} \theta\wedge\mathrm{d} \theta=0$

  • $\text{ker }\mathrm{d}_0=\left\{f\middle|\dfrac{\partial f}{\partial \theta}=0\right\}=\{f|f=cte\in\mathbb{R}\}$
  • $\text{im }\mathrm{d}_{-1}=\{0\}$

$$H^0(\mathbb{S}^1)=\dfrac{\{f=cte\in\mathbb{R}\}}{0}\simeq\mathbb{R}$$


  • $\text{ker }\mathrm{d}_1=\{g\mathrm{d}\theta\}$
  • $\text{im }\mathrm{d}_0=\left\{\dfrac{\partial f}{\partial \theta}\mathrm{d}\theta\right\}$

and we wonder whether every $2\pi$-periodic function may be expressed as the derivative of some other $2\pi$-periodic function $f$. It should be then $f(\theta)=\int_0^\theta g(t)\mathrm{d} t$ (up to constants), but for $f$ to be $2\pi$-periodic it is needed that $\int_0^{2\pi} g(t)\mathrm{d} t=0$. Obviously not every $2\pi$-periodic function $g$ meets this requirement, but one may overcome this by slightly modifying $g$: $\tilde{g}=g-\frac{1}{2\pi}\int_0^{2\pi} g(t)\mathrm{d} t$. This way every $g$ is expressed in a unique way as the sum of a constant function and a function of the type $\dfrac{\partial f}{\partial \theta}$. As a consequence,

$$H^1(\mathbb{S}^1)=\dfrac{\{g\mathrm{d}\theta\}}{\left\{\dfrac{\partial f}{\partial \theta}\mathrm{d}\theta\right\}}\simeq\mathbb{R}$$

In this case, $\mathrm{d}\theta$ acts as a generator of $H^1(\mathbb{S}^1)$. Same to $\mathbb{R}^2-(0,0)$, it is closed (locally related to the angle function $\theta$) and assigns nonzero values to nonzero homological objects, i.e. the circumference loop itself