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.
Best Answer
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 $A$.