Let $R = {\bf Z}$
Let $\partial_i : C_i \rightarrow C_{i-1}$ be a boundary map
where $C_{-1} = \{ 0 \}$, $C_i$ is the set of all maps $f$ and $f: \Delta_i \rightarrow M$. Let $Z_n = $Ker of $\partial_n$
If $f $ is a loop in $M$ then $\partial_1(f)=0$.
If $M$ is $S^2$, and if $f$ is in $Z_2$, what is $f$ ?
I want to know the example for $f$.
In fact, if $f$ maps the boundary of $\Delta_2$ into a point in $M$, i.e., $\Delta_2$ wraps $M $ one time, $f$ is in $Z_2$. However does it imply that $\partial_2 (f) =0$ ?