Homology of closed orientable surfaces

Example 3.3.18 (Orientable surfaces). Let \(M_g\) be the closed orientable surface of genus \(g\). It admits a CW-structure with one \(0\)-cell, \(2g\) one-cells \(a_1, b_1, \ldots , a_g, b_g\), and one \(2\)-cell attached along the word \([a_1, b_1] \cdots [a_g, b_g] = a_1 b_1 a_1^{-1} b_1^{-1} \cdots a_g b_g a_g^{-1} b_g^{-1}\). The cellular chain complex is \[ 0 \to \Z \xrightarrow {d_2} \Z ^{2g} \xrightarrow {d_1} \Z \to 0. \] The boundary \(d_1\) is zero because each one-cell starts and ends at the unique zero-cell. For \(d_2\), each generator \(a_i\) appears in the attaching word with total exponent \(1 + (-1) = 0\), and similarly for \(b_i\). Thus each map \(\Delta _{\alpha \beta }\) has degree zero, giving \(d_2 = 0\). Since both boundary maps vanish, we obtain \(H_0(M_g) \cong \Z \), \(H_1(M_g) \cong \Z ^{2g}\), and \(H_2(M_g) \cong \Z \).