Let $l_1$ , $l_2$ and $l_3$ be three paths in X with $l_1 (0) = l_3 (1)$, $l_1 (1) = l_2 (0)$ and $l_2 (1) = l_3 (0)$. Define the loop $l = l_1 \cdot l_2 \cdot l_3 $ (based at $l_1 (0)$).
Show that $l$ and $l_1 + l_2 + l_3$ are 1-cycles. (This can just be shown by applying the boundary map, $\partial$, on the paths are to get $0$. Hence they are in $\ker(\partial)$, which defines them as 1-cycles).
Furthermore, show that $[l] = [l_1 + l_2 + l_3] \in H_1(X)$
My confusion is regarding treating this as an algebraic problem, or proving it through the definition of homotopy equivalence, i.e using the definition of the class $[l]$ as everything homotopy equivalent to the path $l$