- Show that \(\chi(\Sigma_g + \Sigma_h) = \chi(\Sigma_g) + \chi(\Sigma_h) - 2\).
Solution
1. For closed orientable surfaces, \(\chi(\Sigma_g)=2-2g\).
Use the standard CW structure with one \(0\)-cell, \(2g\) \(1\)-cells, and one \(2\)-cell.
2. The connected sum satisfies \(\Sigma_g\#\Sigma_h\cong\Sigma_{g+h}\).
Connected sum adds the numbers of handles.
3. Hence\[\boxed{\chi(\Sigma_g\#\Sigma_h)=\chi(\Sigma_g)+\chi(\Sigma_h)-2.}\]
The left side is \(2-2(g+h)\), while the right side is \((2-2g)+(2-2h)-2=2-2g-2h\).