let $X$ be topological space and let $C_{n}(X), n \in \mathbb{Z}$ denote singular chain complex. I don't at all understand how and when can we claim a class $[\delta]\in H_{n}(X)$ to be its generator.
For example when $X$ is a torus, I know that $H_{1}(X)=\mathbb{Z} \oplus \mathbb{Z} $ but I don't understand why can we say that red and blue loops (as pictured below) generate it. How do I know they are "independent" as well? How do I know they don't belong in a boundary?
Consider even simpler example, the sphere $S^1$. How do I know which class $[\delta]$, where $\delta:\Delta^1 \to S^1 \in Z_{1}(S^1)$, generate $H_{1}(S^1)$ ?
Thanks in advance.
To show the red loop $\gamma$ is not a boundary on the torus $T$, you could show that there is a closed $1$-form $\omega$ on $T$ such that $$\int_\gamma\omega\ne0.$$
Represent $T$ as $\Bbb R^2/\Bbb Z^2$ and $\gamma$ as the path $[0,1]\to T$ defined by $\gamma T=(t,0)$. This is a closed path: $(0,0)=(1,0)$ in $\Bbb R^2/ \Bbb Z^2$. The $1$-form $\omega=dx$ is closed and $$\int_\gamma\omega=\int_0^1dt=1\ne0.$$
For the torus $T$ a loop represents the trivial homology class iff $$\int_\gamma dx=\int_\gamma dy=0.$$ This is equivalent to it lifting from a loop in $\Bbb R^2/\Bbb T^2$ to a loop in $\Bbb R^2$.