Cellular homology of a two-dimensional CW complex

For a two-dimensional CW complex $X$, the cellular chain complex $(C_i^c(X), \partial_i)$, $i = 0, 1, 2$, is defined as follows (with all higher $H_i^c(X) = 0$, since there are no cells in those dimensions). Let $C_0^c(X)$ be the free abelian group with one generator $g_i^0$ for each $0$-cell $e_i^0$, and set $\partial_0 = 0$. For each $1$-cell $e_i^1$, choose the standard positive orientation of the interval, giving a generator $g_i^1$ for this oriented $1$-cell; if it is attached via $f$ so that $f(1) = e_j^0$, $f(0) = e_k^0$, then $\partial_1(g_i^1) = g_j^0 - g_k^0$. For each $2$-cell $e_j^2$, choose the standard positive orientation of the cell, giving a generator $g_j^2$; when the $2$-cell is attached via $f_j : S_j^1 \to X^1$, the induced map on $H_1$ determines the coefficients $n_{ij}$ of $\partial_2(g_j^2) = \sum_i n_{ij} g_i^1$ by computing the degree of the composition of $f_j$ with the map collapsing all of $X^1$ except the $1$-cell $e_i^1$ to a point. The cellular homology groups are then $H_i^c(X) = \ker (\partial_i)/\mathrm{im}(\partial_{i+1})$.

For example, for the torus $T$, the $2$-cell is attached to $X^1$ via the commutator $aba^{-1}b^{-1}$, so $\partial_2(g^2) = g_a^1 + g_b^1 - g_a^1 - g_b^1 = 0$; since there is only one $0$-cell, $\partial_1(g_a^1) = \partial_1(g_b^1) = 0$ as well. Thus all boundary maps are $0$, giving $H_2^c(T) \simeq \mathbb {Z}$, $H_1^c(T) \simeq \mathbb {Z} \oplus \mathbb {Z}$, $H_0^c(T) \simeq \mathbb {Z}$. For the Klein bottle, the attaching map of the $2$-cell instead follows the pattern $abab^{-1}$, giving $\partial_2(g^2) = g_a^1 + g_b^1 + g_a^1 - g_b^1 = 2g_a^1$; thus $\ker (\partial_2) = 0$, so $H_2^c = 0$, while $H_1^c(X) = \mathbb {Z}(g_a^1) \oplus \mathbb {Z}(g_b^1)/\left\{ 2g_a^1\right\} \simeq \mathbb {Z}_2 \oplus \mathbb {Z}$, and again $H_0^c(X) \simeq \mathbb {Z}$.

See Exercise 5.3.3.