CW complex that deformation retracts to annulus and Möbius band

Question

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.

Answer 1

Here is a cell complex $X$ containing subcomplexes $A\cong M$ (the Möbius strip) and $B\cong \mathbb{S}^1\times I$ such that $X$ deformation retracts to both $A$ and $B$ (note that several vertices and edges are identified):

                                  

The subcomplex $A$ is

                                 

which, when the appropriate points and edges are identified, looks like

                           

which deformation retracts onto $C=(p_2\cup e_5)\cong\mathbb{S}^1$. The subcomplex $B$ is

                          

which, when the appropriate points and edges are identified, looks like

                           

which deformation retracts onto $C=(p_2\cup e_5)\cong\mathbb{S}^1$.

Note that $A\cap B=C$. We can extend the deformation retraction of $A$ onto $C$ to all of $X$ by the identity on $B$, so that $X$ deformation retracts onto $B$, and similarly with the deformation retraction of $B$ onto $C$, so that $X$ deformation retracts onto $A$.

Answer 2

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.)

Answer 3

We have a map, $M\to S^1$ that contracts $M$ to it's central circle. When we take the mapping cylinder, one end is a circle. Then attach a cylinder to that circle.