I am a physics student, and I am having trouble understanding the relationship between Laplacian and Euler numbers through concrete calculations. In my understanding, the "dimension" of the scalar field satisfying $\Delta_{(0)} \phi=0$ corresponds to the $0$th betti number, and the "dimension" of the vector field satisfying $\Delta_{(1)} u^{\mu}=0$ corresponds to the $1$st betti number. For simplicity, Let me consider $S^2$ with radius $1$. The Laplacian for scalar is then \begin{align} \Delta=\frac{1}{\sin \theta}\frac{\partial}{\partial\theta}(\sin \theta\frac{\partial}{\partial\theta})+\frac{1}{\sin^2 \theta}\frac{\partial^2}{\partial\varphi^2} \end{align} and the solution of $\Delta \phi=0$, \begin{align} \phi=e^{im\varphi}f(\theta)\ \ {\rm and}\ \ &f(\theta)\ {\rm satisfies}\ \frac{\partial}{\partial\theta}(\sin \theta\frac{\partial}{\partial\theta}f(\theta))-m^2f(\theta)=0 \end{align} The three questions are as follows.
(i) Sphere has $0$th betti number $1$. Which dimension does the above solution refer to? In other words, the above solution contains parameters such as $m$. Does it mean the number of such parameters?
(ii) I know the solution of vector-Laplacian. it's just $u_{\mu}=\nabla_{\mu} \phi$ and $u_{\mu}=\epsilon_{\mu\nu}\nabla^{\nu}\phi$. The reason is $\Delta_{(1)}\nabla_{\mu}\phi=\nabla_{\mu}(\Delta_{(0)}\phi)=0$. Thus, the number of solutions is twice as large as the number of solutions of the scalar field. However, sphere has $1$st betti number $0$. What additional conditions preclude these solutions?
(iii) $S^2$ is not bounded, but if it exists, what kind of change does the betti number undergo? Also, is there an equivalent Neumann boundary condition for scalar fields in vector fields?
Please tell me about the above. Please note that I am a physics student, so I would like to check the differential equations by writing them down instead of the differential forms.