Homology of closed non-orientable surfaces

Example 3.3.19 (Non-orientable surfaces). Let \(N_g\) be the closed non-orientable surface of genus \(g\) (the connected sum of \(g\) copies of \(\RP ^2\)). It admits a CW-structure with one \(0\)-cell, \(g\) one-cells \(a_1, \ldots , a_g\), and one \(2\)-cell attached along the word \(a_1^2 a_2^2 \cdots a_g^2\). The cellular chain complex is \[ 0 \to \Z \xrightarrow {d_2} \Z ^{g} \xrightarrow {d_1} \Z \to 0. \] Again \(d_1 = 0\), since each one-cell starts and ends at the unique zero-cell. For \(d_2\), each \(a_i\) appears with total exponent \(2\), so the degree of each \(\Delta _{\alpha \beta }\) equals \(2\). Thus \(d_2(1) = (2, 2, \ldots , 2) \in \Z ^g\). Since \(d_2\) is injective, \(H_2(N_g) = 0\). To compute \(H_1(N_g) = \ker (d_1)/\im (d_2) = \Z ^g / \langle (2, \ldots , 2) \rangle \), we change basis: replacing the last standard basis vector \((0, \ldots , 0, 1)\) by \((1, \ldots , 1)\), we see that \(H_1(N_g) \cong \Z ^{g-1} \oplus \Z /2\Z \).