Let $X$ be a connected CW-complex. It's easy to see that the 1-cycles (of the cellular chain complex) $\sigma=\sum_i \varepsilon_i\sigma_i\;(\varepsilon_i=\pm 1)$ are linear combinations of 1-cells $\sigma_i$ that (considered as paths with direction given by $\varepsilon_i$) can be composed such that the composition is a loop.

**Question:** Is there also a nice description of the 1-boundaries $B_1$ that allows to show $Z_1/B_1\cong \Pi_1(X)_{ab}$ ?

According to the cellular boundary formula (cf. Hatcher, p. 140), the boundary operator of a 2-cell is the linear combination of whose 1-cells that form the boundary of the 2-cell, weighted with some mapping degree of the attaching map. But from this description I find it impossible to see $Z_1/B_1 \cong \Pi_1(X)_{ab}$.