I'm trying to prove that
$f: X \rightarrow Y$ is a homotopy equivalence $\iff$ $X$, $Y$ are both homeomorphic to a deformation retract of a space $Z$
The $\Leftarrow$ was not a problem. If both are deformation retracts it follows that $Z \simeq X$ and $Z \simeq Y$ and by transitivity $X \simeq Y$.
The first half of $\Rightarrow$ was also not a problem, because using the mapping cylinder $Z_f$ as their common super-space, $X \times I$ can be deformation-retracted down to $Y$ by the homotopy $(x,s,t)\mapsto (x,st)$, i.e. along a vertical line.
Now I'm stuck with the deformation retraction of $Z_f$ onto $X$. For example, do I construct a homotopy from $Z_f$ to $X \times I$ and then to $X$ or directly from $Z_f$ to $X$? I'm also not sure where to use $f \circ g \simeq id_Y$ and $g \circ f \simeq id_X$ and $id_{Z_f} \simeq i \circ r_Y$, where $r_Y$ is the retraction from $Z_f$ onto $Y$.
I'm tempted to do something like this but it doesn't seem to lead anywhere: $i \circ r_Y \simeq id_Y \simeq f \circ g$.
Can someone help me finish this proof? I'd much appreciate any help!
Best Answer
The key point is that the inclusion of $X$ into the mapping cylinder is a cofibration. It is a general fact that if $A \hookrightarrow B$ is a cofibration and a homotopy equivalence, then $A$ is a deformation retract of $B$. One way to think of this is to use the model structure on topological spaces (due to Strom) where (closed) Hurewicz cofibrations are cofibrations, Hurewicz fibrations are fibrations, and weak equivalences are homotopy equivalences.
So let's prove the following more general fact. Let $A \hookrightarrow B$ be an acyclic cofibration in a (closed) model category, where $B$ is fibrant. Then $B$ "deformation retracts" onto $A$. Let us assume that our model categories have functorial factorizations, so there is always a functorial cylinder object $A \times I$ for $A$.
To see this, let us first show that $B $ retracts onto $A$. By replacing the model category with the model category of objects under $A$, we can assume that $A$ is the initial object $\emptyset$, and $B$ is cofibrant in such a way that $\emptyset \to B$ is a weak equivalence. This we can do by considering the lifting diagram with $\emptyset \to B$ and $\emptyset \to \ast$ (where $\ast$ is the final object). We get a retraction $B \to \emptyset$.
Now we want a "deformation retraction." Let us consider the two maps $B \rightrightarrows B$ given by the identity and the retraction. We want a homotopy between the two. But we can consider the lifting diagram with $B \sqcup B \to B \times I$ (for $B \times I$ a cylinder object) and $B \to \ast$. Since $\emptyset \to B$ is an acyclic cofibration, so is $\emptyset \to B \sqcup B$, and two-out-of-three shows that $\emptyset \to B \times I$ is a weak equivalence too; thus $B \sqcup B \sqcup \emptyset \times I \to B \times I$ is a trivial cofibration. Thus a lifting exists in the diagram, which is the map $B \times I \to B$ that we wanted.