TITLE: CW complex that deformation retracts to annulus and Möbius band
QUESTION [5 upvotes]: I am trying to construct a 2-dimensional CW complex that contains both an annulus, thought of as $S_{1}\times I$ and a Möbius band as deformation retracts.
The first part of the problem asked to show that the mapping cylinder of every map $f:S^{1}\rightarrow S^{1}$ is a CW complex so I thought of doing the construction using mapping cylinders, perhaps gluing the mapping cylinder of one map to the mapping cylinder of another map along the base circle. One map would be the identity map on $S^{1}$ since its mapping cylinder will be $S^{1}\times I$, but I can't figure out what the other map should be in order to get a Möbius band.

REPLY [2 votes]: In general, if $W = X \bigsqcup_{A} Y$ is the pushout of two CW complexes along a common subcomplex $A$, $X$ deformation retracts onto $A$, and $Y$ deformation retracts onto $A$, then $W$ deformation retracts onto both $X$ and $Y$, where the retracts from $Y$ and $X$ to $A$ are used to retract $W$ to $X$ and $Y$, respectively. (Hatcher, Prop 0.19 and Cor 0.20.)
To use this to give intuition for the construction without trying to visualize it with pictures (although pictures are great!!), take $X$ to be the annulus and $Y$ to be the Möbius band. Place CW structures on each of these so there is a canonical $S^1$ that is a deformation retract of each, and then just identify the two CW complexes by this $S^1$, where the orientation and vertices match. This will yield the CW complex that has both $X$ and $Y$ as deformation retracts.
(IMO, Hatcher's part (a) is just a red herring for solving this problem.)