# Hatcher 0.17

Hatcher, Algebraic Topology, Chapter 0

17. (a) Show that a mapping cylinder of every map $f:S^1\to S^1$ is a CW complex.

(b) Construct a 2-dimensional CW complex that contains both an annulus $S^1\times I$ and a Möbius band as deformation retracts.

Proof. (a) Let $f:S^1\to S^1$ be arbitrary and consider the mapping cylinder $S^1\sqcup_{f} S^1$. In particular, the space $X$ “looks like” a “hollow cylinder” $S^1\times I$ attached to a copy of $S^1$ by the identification $(x,1)\sim f(x)$ for all $x$ in the “top copy” $S^1\times\{0\}$ of $S^1$. The process to construct a CW structure on $X$ is as follows:

1. Begin with two points $x_0=e^0_1$ and $f(x_0)=e^0_2$.
2. Construct the 1-skeleton $X^1$ of $X$ by adding three 1-cells $e^1_i$, $i=1,2,3$:
• Add two 1-cells $e^1_1$ and $e^1_2$ as disjoint circles for which $e^0_i\in e^1_i$, $i=1,2$.
• Let $e^1_3$ be the 1-cell formed from the “graph” of $f$, drawn “around” the “cylindrical part” of $S^1\times I$.
3. Attach a 2-cell $e^2_1\cong D^2$ by the attachment map that identifies $\partial e^2_1\cong S^1$ with the 1-skeleton $X^1$ formed above.

Figure 1 below shows a (very crude) illustration of what such a thing might look like. Note that the red curve in Figure 1 represents the “graph” of $f$, where the thickened red line represents the part of the graph “on the front” of the cylinder and the thinner line represents the part “on the back” of the cylinder.

Figure 1

(b) This problem follows a pattern similar to that in part (a). First, let $M$ denote a Möbius band and let $A$ denote an annulus. Here, note that both $M$ and $A$ retract onto their center circles $S^1_M$ and $S^1_A$, respectively. Let $f:M\to S^1$ and $g:A\to S^1$ denote these retractions.

Let $X_1$ and $X_2$ be defined to be the mapping cylinders associated with $f$ and $g$, respectively; that is,

$X_1=S^1\sqcup_f M$ and $X_2=S^1\sqcup_g A$.

Note, in particular, that both spaces $X_i$, $i=1,2$, deformation retract onto their “range” subspaces as noted on page 2 of the text, i.e., $X_1$ deformation retracts onto $M$ and $X_2$ deformation retracts onto $A$. Finally, one can form the desired space $X$ by “gluing together” the spaces $X_i$, $i=1,2$, via the identity map on $S^1$. Visually, then, $X$ looks like a paper towel roll with a Möbius band $M$ glued to one end and an annulus $A$ glued to the other, and by previous remarks, $X$ deformation retracts onto the spaces $M$ and $A$.   $\square$