Exercise 0.1

Proof. Using the CW complex construction of the torus on p. 5, we have the map denoted by the small arrows:

To prove this map is indeed a deformation retraction, we use the identification of the unit square with the unit disc (in this case the boundary of the circle is divided into four arcs with the labelling scheme $\mathit{ab}{a}^{-1}{b}^{-1}$), and using polar coordinates we can let $\tilde{F}((r,𝜃),t)=(r+t(1-r),𝜃)$. Then, let $F=q\circ \tilde{F}$ where $q$ is the quotient map shown as the last arrow above. $F$ is continuous as $\tilde{F}$ is continuous in each coordinate, and then $q$ is continuous. $F$ is also such that $F((r,𝜃),0)=(r,𝜃)$, $F((r,𝜃),1)=(1,𝜃)$, and $F((1,𝜃),t)=(1,𝜃)$. Thus, $F$ is a deformation retraction. The two circles in the last diagram are the longitude and meridian circles of the torus by construction, namely since in the penultimate diagram we identify the sides marked $a$ to get the meridian circle and then the sides marked $b$ to get the longitude circle. □