Example 25.3 (The torus and the Klein's bottle):

We now show that the Klein's bottle and the torus are obtained by attaching a two cell to the figure eight space $S^1\vee S^1$. In both cases we take $X = I^2$ to be the two cell, $A = \dot{I}^2$ the boundary of $I^2$ and $B = S^1\vee S^1$ regarded as a subset of $S^1\times S^1$ namely $(S^1\times\{1\})\cup (\{1\}\times S^1)$. The distinguishing factor is that the attaching map $f:A\longrightarrow B$ is different in the two cases.

1. For the torus we define $f:A\longrightarrow B$ to be the continuous surjection
   

   $$f(x, 1) = f(x, 0) = (e^{2\pi ix}, 1),\quad x \in [0,1]$$

   $$f(1,y) = f(0, y) = (1, e^{2\pi iy}),\quad y\in[0,1]$$

   
   It is geometrically clear that $X\sqcup_f B$ is a torus but we demonstrate this formally owing to the importance of the type of argument involved. Let $p:I^2\longrightarrow S^1\times S^1$ be the quotient map, $i_X:X\longrightarrow X\sqcup B$ the inclusion map and $\eta:X\sqcup B\longrightarrow X\sqcup_fB$ the quotient map. The map
   $$\phi:S^1\times S^1 \longrightarrow X\sqcup_fB$$
   given by $\phi(\exp(2\pi ix), \exp(2\pi iy)) = (\eta\circ i_{X})(x, y)$ is well-defined, bijective and its continuity follows from the fact that $\phi\circ p = i_X\circ \eta$ and $i_{X}\circ \eta$ is continuous. Finally the compactness of $S^1\times S^1$ and the fact that $X\sqcup_f B$ is Hausdorff shows that $\phi$ is a homeomorphism.
2. The argument for the Klein's bottle proceeds along similar lines and we merely indicate the attaching map $f:I^2\longrightarrow S^1\vee S^1$ namely,
   

   $$f(x, 1) = f(x, 0) = (e^{2\pi ix}, 1),\quad x \in [0,1]$$

   $$f(1,y) = f(0, 1-y) = (1, e^{2\pi iy}),\quad y\in[0,1].$$

   
   
3. It is sometimes convenient to take the closed unit disc $E^2$ as the two cell. But the attaching map $f:S^1\longrightarrow S^1\vee S^1$ would be slightly more complicated to write down. For the Klein's bottle the attaching map is given by
   $$f(z) = \left\{\begin{array}{lll} (z^4, 1) & & 0 \leq \mbox{arg}\... ...}) & & 3\pi/2 \leq \mbox{arg}\;z \leq 2\pi \\ \end{array} \right. \eqno(25.5)$$
   For the torus the attaching map is obtained from (25.5) by suppressing the negative signs in the last two expressions. The student is invited to work out a similar construction for the double torus as well.