# Hatcher 0.4

Hatcher, Algebraic Topology, Chapter 0

4. A deformation retraction in the weak sense of a space $X$ to a subspace $A$ is a homotopy $f_t:X\to X$ such that $f_0=1_X$, $f_1(X)\subset A$, and $f_t(A)\subset A$ for all $t$. Show that if $X$ deformation retracts to $A$ in this weak sense, then the inclusion $A\hookrightarrow X$ is a homotopy equivalence.

Proof. For preliminary information regarding the definitions of homotopy equivalences, homotopy inverses, etc., see previous results.

Let $X$ be a topological space, let $A\subset X$ be a subset, and suppose that $X$ deformation retracts weakly onto $A$ by way of the homotopy $f_t:X\to X$. Let $\iota:A\to X$ denote inclusion. It suffices to show that $\iota$ is a homotopy equivalence, i.e. that there exists a map $g:X\to A$ for which $\iota g\simeq 1_X$ and $g\iota\simeq 1_A$. Note that $f_1(X)\subset A$, and so while $f_1:X\to X$, it’s natural to consider $f_1$ as a map $X\to A$. For that reason, the obvious candidate for the map $g$ mentioned above is

$g(x)\overset{\text{def}}{=}f_1(x)$ for all $x\in X$.

It suffices to show that the this choice of $g$ has the desired properties.

By hypothesis, there is a homotopy $f_t:X\to X$. Let $F:X\times I\to X$ denote this homotopy (that is, let $f_t(x)=F(x,t)$ for all $x\in X,t\in I$). It follows, then, that $F(x,0)=id_X$ and that $F(x,1)=g(x)$. In particular, because $f_1(X)=g(X)\subset A$, $\iota\circ g=g$ for all $x\in X$. Thus,

$F(x,0)=id_X$ and $F(x,1)=f_1(x)=g(x)=\iota\circ g(x)$ for all $x\in X$,

and so $F$ is a homotopy connecting $f_0=id_X$ to $g=\iota\circ g$. Hence, it follows that $\iota\circ g\simeq id_X$.

Next, define a map $G:A\times I\to A$ by $G(a,t)=F(a,t)=f_t(a)$ for all $a\in A$, where $F$ is the homotopy above. Then $G(a,0)=F(a,0)=f_0(a)$, i.e. $G(\cdot,0)=f_0|_A=id_X|_A=id_A$. Similarly, $latex$G(a,1)=F(a,1)=f_1(a)=g(a)=(g\circ\iota)(a)\$ for all $a\in A$. Hence, $G$ is a homotopy connecting $g\circ\iota$ to $id_A$, whereby it follows that $g\circ\iota\simeq id_A$. This, combined with the result above, verifies that $\iota$ is a homotopy equivalence with homotopy inverse $g=f_1$.   $\square$