I am trying to understand and "visualize" the generator of the first de Rham cohomology group of the circle and the intuition behind the concept of angular form.
$H^1(S^1)$ can be computed from Mayer-Vietoris sequence obtaining $\dim H^1(S^1)= 1$ and from the coboundary map we obtain a generator $\omega_{MV}$ of $H^1(S^1)$ as a bump form with support (in a connected component of) the intersection of the two open sets used in M-V, that is, a bump form on the circle (say with integral equal to 1). Thus $H^1(S^1) = \langle [\omega_{MV}] \rangle$.
This form is sometimes called "angular form" but intuitively does not look like a "differential" of an angle (I know that the angle is a multifunction and hence we should use the universal cover of $S^1$ to properly discuss it, but we can get away with some careful intuition nonetheless). Intuitively the angle "function" has a linear growth on the circle minus a point, thus I'd expect its "differential" to be a "constant" form.
Of course constant does not make sense for a 1-form. We can restate this by asking that this angular form should evaluate to a constant function when paired with the vector field $X = x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}$ restricted to $S^1$. Of course such a form can be seen as the restriction of $xdy - ydx$ to $S^1$.
Without referring to the ambient space, we can construct a "constant" form $\omega_c$ gluing the forms $c\cdot dx$ on $U,V \simeq \mathbb{R}$ as follows. Say $U \simeq_{\varphi_U} (0-\varepsilon,\pi +\varepsilon)$ and $V \simeq_{\varphi_V} (\pi-\varepsilon,2\pi +\varepsilon)$ just to fix the ideas. We have $$\varphi_U(U\cap V) = (0- \varepsilon,0+\varepsilon) \sqcup (\pi -\varepsilon,\pi+\varepsilon)$$ $$\varphi_V(U\cap V) = (2\pi- \varepsilon,2\pi +\varepsilon) \sqcup (\pi -\varepsilon,\pi+\varepsilon)$$ and the transition functions are the identity on the first compoment and the translation $\pm 2\pi$ on the second component, thus their jacobian is always the identity. Moreover on $\mathbb{R}$ the forms $dx$ are translation invariant, i.e. $dx_p = dx_{p+q}$, thus in the end $\varphi_{UV}^*dx = dx$. This defines $\omega_c$ as $\omega_c|_U = c \cdot \varphi_U^*dx$ and $\omega_c|_V = c \cdot \varphi_U^*dx$.
These forms are closed by definition and they are intuitively exact since if there was a function on $S^1$ with a constant differential, it must be constantly increasing, but this is impossible on $S^1$. Thus $H^1(S^1) = \langle [\omega_{c}] \rangle$.
This form is what I would call angular form, of course, since $H^1$ is 1-dimensional, $[\omega_{c}] = [\omega_{MV}]$ thus by extension we call also $\omega_{MV}$ an angular form.
Since $S^1$ is compact, we also have $H_c^1(S^1) = \langle [\omega_{c}] \rangle = \langle [\omega_{MV}] \rangle$.
Does my reasoning make sense?
Thanks
EDIT: I've rewritten the construction of "constant" 1-forms to make it clearer. Seems to me it that the use of partition of unity it not even necessary.