Relation between closed 1-chain and closed paths Let $\gamma\in Z_1X=\mathrm{kernal}\,\partial_{1}$, i.e., a closed 1-chain. Prove there exists a 1-chain $\delta=\sum_in_i\delta_i$, where each $\delta_i$ is a closed path, such that $\gamma-\delta\in B_1X=\mathrm{image}\,\partial_2=$ constant 1-chains, i.e., $\gamma$ is homologous to $\delta$.
How do I prove this? Can we easily construct those closed paths $\delta_i$ from the components of $\gamma$?
Also, is there a generalization of this for each $n$-chains?
 A: This follows from the Hurewicz theorem, which includes the statement that for path-connected $X$ the natural map
$$
\pi_1(X,x)\to H_1(X)
$$ 
is surjective. This means that every 1-cycle in $X$ is homologous to a loop in $X$. If $X$ is not connected, apply this theorem to each path-connected component of $X$. 
A: I found a solution to this problem:
First, write $\gamma=\sum_in_i\gamma_i$. We may assume all $n_i>0$ by reversing some of the $\gamma_i$. Next, we do induction on $n=\sum_in_i$. For $n=1$, the result is trivial. For $n>1$, $\gamma_1(1)=\gamma_k(0)$ for some $k$. If $k=1$, then $\gamma_1$ is already a closed path. Then the rest $\gamma-\gamma_1$ has less sum of all coefficients. By induction, it can be written as a sum of closed paths. If $k>1$, let $\delta=\gamma_1*\gamma_k$, which is homologous to $\gamma_1+\gamma_k$. Then $\gamma'=\delta+(n_1-1)\gamma_1+(n_k-1)\gamma_k+\sum_{i\neq1,k}n_i\gamma_i$, which is homologous to $\gamma$. Again, the sum of coefficients of $\gamma'$ is $(\sum_in_i)-1$. By induction, $\gamma'$ can be written as a sum of closed paths.
