Exercise 0.17

Solution. Consider the CW complex and the maps drawn with small arrows below:

In the top row, since the segment labelled $c$ is homeomorphic to $[0,1]$, we see that if the unit square containing $c$ has the coordinate along $c$ as the $x$ coordinate, and the other edge as the $y$ coordinate, our map is given by $F((x,y),t)=((1-t)x,y)$ in this unit square, and the identity on the unit square containing $a,b$. $F$ is well-defined since it acts the same on the identified edges. $F$ is continuous in each coordinate hence continuous and is such that $F((x,y),0)=(x,y)$, $F((x,y),1)=(0,y)$, and $F((0,y),t)=(0,y)$; thus, $F$ on the entire complex is continuous by the pasting lemma. $F$ is therefore a deformation retract to the Möbius strip defined by the unit square containing $a,b$.

In the bottom row, let the horizontal edges be the $x$ coordinate axis with $x\in [0,1]$, and the vertical edges as the $y$ coordinate. Then, our map is given by $F((x,y),t)=(x+t(1∕2-x),y)$ in this unit square, and the identity on the unit square containing $c$. $F$ is well-defined since it acts the same on the identified edges. $F$ is continuous in each coordinate hence continuous and is such that $F((x,y),0)=(x,y)$, $F((x,y),1)=(0,y)$, and $F((1∕2,y),t)=(1∕2,y)$; thus, $F$ on the entire complex is continuous by the pasting lemma. $F$ is therefore a deformation retract to the annulus defined by the unit square containing $c$. □