Hatcher 0.9

Hatcher, Algebraic Topology, Chapter 0

9. Show that a retract of a contractible space is contractible.

Proof. Let X be a topological space which is contractible, A\subset X a subspace, and suppose that X retracts onto A via the map r. To be more precise, suppose x is a point and that f:X\to \{x\} is a map with homotopy inverse g:\{x\}\to X for which fg\simeq 1_{\{x\}} and gf\simeq 1_X. Let F:\{x\}\times I\to \{x\} (resp. G:X\times I\to X) denote the homotopy connecting fg and 1_\{x\} (resp., connecting gf and 1_X). It suffices to find homotopies connecting A to \{x\} and vice versa.

Note that r:X\to A being a retract implies immediately that the inclusion \iota:A\to X satisfies r\iota=\iota r= 1_A. Solution of this problem, then, boils down to finding maps h:A\to\{x\} and k:\{x\}\to A so that the diagram in Figure 1 below commutes and for which hk\simeq 1_A, kh\simeq 1_{\{x\}}.

Hatcher 0.9
Figure 1

The obvious candidates here are h=f\iota and k=rg, which by definition ensure that the diagram in Figure 1 commutes. Moreover,

hk = (f\iota)(rg)=f(\iota r)g=fg\simeq 1_{\{x\}} and kh=(rg)(f\iota)=r(gf)\iota\simeq r\iota=1_A,

whereby it follows that h and k satisfy the desired properties with regards to the identity maps. Hence, the claim is proven.    \square

Note: The choices for h and k can be rectified with the “usual homotopy notation” by modifying the definitions of F and G given above simply by working relative to A throughout. To be specific, let F:\{x\}\times I\to\{x\} and G:X\times I\to X be the homotopies mentioned above sending fg\mapsto id_{\{x\}} and gf\mapsto id_X, respectively. In particular, F(y,0)=(f\circ g)(y), F(y,1)=x, G(y,0)=(g\circ f)(y), and G(y,1)=y for all y\in X. Next, define homotopies H:\{x\}\times I\to\{x\} and K:A\times I\to A so that H(y,t)=F(y,t)|_A and K(y,t)=G(y,t)|_A. It follows immediately, then, that H(y,0)=F(y,0)=(f\circ g)(y), H(y,1)=F(y,1)=x, K(y,0)=G(y,0)=(g\circ f)(y), and K(y,1)=G(y,1)=y for all y\in X\cap A=A. Thus, A has the same homotopy type as \{x\} by way of the explicit homotopies H and K.   \square

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