Let \(X\) be a space obtained by pasting the edges of a polygonal region together in pairs.
Show that \(X\) is homeomorphic to exactly one of the spaces in the following list: \(S^2\), \(P^2\), \(K\), \(T_n\), \(T_n \# P^2\), \(T_n \# K\), where \(K\) is the Klein bottle and \(n \geq 1\).
Show that \(X\) is homeomorphic to exactly one of the spaces in the following list: \(S^2\), \(T_n\), \(P^2\), \(K_m\), \(P^2 \# K_m\), where \(K_m\) is the \(m\)-fold connected sum of \(K\) with itself and \(m \geq 1\).
Solution
By Theorem 77.5, every such quotient is homeomorphic to exactly one of \[ S^2,\qquad T_g\ (g\ge1),\qquad P_r\ (r\ge1), \] where \(T_g\) is the orientable surface of genus \(g\) and \(P_r\) is the connected sum of \(r\) projective planes. Uniqueness of these standard surfaces follows from orientability and Euler characteristic: \[ \chi(T_g)=2-2g,\qquad \chi(P_r)=2-r, \] and \(S^2\) is the unique orientable case with Euler characteristic \(2\).
We have \[ P_1=P^2,\qquad P_2\cong K. \] Lemma 77.4 converts one torus summand together with one crosscap pair into two additional crosscaps; equivalently \[ T_1\# P_r\cong P_{r+2}. \] Inductively, \[ P_{2g+1}\cong T_g\# P^2, \qquad P_{2g+2}\cong T_g\# K. \] Thus the standard list \(S^2,T_g,P_r\) can be rewritten exactly as \[ S^2,\quad P^2,\quad K,\quad T_g,\quad T_g\#P^2,\quad T_g\#K \qquad(g\ge1). \] Since the original standard forms are unique, exactly one member of this list represents \(X\).
Put \(K_m=K\#\cdots\#K\) with \(m\) factors. Since \(K\cong P_2\), \[ K_m\cong P_{2m}, \qquad P^2\#K_m\cong P_{2m+1}. \] Therefore the same classification can be rewritten as \[ S^2,\quad T_g,\quad P^2,\quad K_m,\quad P^2\#K_m \qquad(g,m\ge1). \] Again these are pairwise distinguished by orientability and Euler characteristic, so exactly one occurs.