Let $M$ be a compact Riemannian manifold without boundary.
a) If $M$ is a sphere, prove that the cohomology space of order $1$ is trivial: $H ^1 (M, \Bbb R) = 0$.
b) If $\omega = \delta\theta$ is the differential $1$-form angle differential on the unit circumference $S ^ 1$, show that $\omega$ is harmonic.
c) If $M = S^1 \times S^1$ (Torus topology) show that the dimension of $H ^ 1 (M, \Bbb R)$ is at least $2$.
a) Using induction and Mayer-Vietoris sequence one can easily check that
$H^{p}_{\mathrm{dR}}(S^n)=\begin{cases}\mathbb{R},\ p\in\{0,n\}\\ 0, \ \mbox{otherwise}\end{cases}$
(Consider open cover of $S^n$ given be sets $U_{\pm}=S^n\setminus\{(0,...,\pm 1)\}$ ).
b) You need to check that $(d\delta +\delta d)\omega =0$. Use the relation between $\delta$ and $\ast d\ast$.
c) It follows from Kunneth formula for cohomology.
