I am new to homology and chains and such, so I am wondering whether my example is correct.
I have seen this question, but it seemed rather complicated, and I wanted to find a simple example without having to look at commutators and such.
Find an example of two closed curves $v$ and $w$, which are homologous (when regarded as 1-cycles), but are not homotopic.
The hint was to consider a surface, so I came up with the following.
Consider $X=\mathbb{C}\setminus\{\pm 1\}$ and consider the loops $$\gamma_\pm:[0,1]\rightarrow X,\quad \gamma_\pm(t)=\pm(e^{2\pi it}-1).$$ Then $\gamma_\pm$ are closed paths starting and ending at $0$ and not homotopic.
Furthermore, regarding $\gamma_\pm$ and 1-chains, we have $$(\gamma_+-\gamma_-)(t) =(e^{2\pi it}-1)-(e^{2\pi it}-1) =0.$$ Thus $\gamma_\pm$ are homologous.
You actually do not regard them as $1$-chains, but as functions $[0, 1] \rightarrow \mathbb C$ and use the additive structure on $\mathbb C$ to subtract them from each other. This is not what happens in homology: You add up their equivalence class in $H_1(X)$ by checking whether they bound a sum of $2$-chains.
The following example is pretty standard. Let $X = \Sigma_2$ (a two-holed torus). Then the curve around the "belly" of $X$ is null-homologous but not null-homotopic. You can probably convince yourself that it is not null-homotopic. It is null-homologous because it bounds a torus with a disk removed, which is a sum of $2$-chains (do you see why?).