# Suppose $X$ is a contractible space and $A \subset X$ with $A$ being homotopy equivalent to $S^2$. Prove that there is no retraction $X \rightarrow A$

Suppose $$X$$ is a contractible space and $$A \subset X$$ with $$A$$ being homotopy equivalent to $$S^2$$. Prove that there is no retraction $$X \rightarrow A$$

Suppose there is a retraction $$r: X \rightarrow A$$. Then $$ri = Id_A$$.

Let's look at the homomorphism induced on the second homology.

$$H_2(A)\stackrel{i_*}{\rightarrow}H_2(X) \stackrel{r_*}{\rightarrow} H_2(A)$$

Since $$A \cong S^2$$ we have $$H_2(A) = \mathbb{Z}$$. Furthermore, $$ri_*=Id_{A_*}$$.

But $$X$$ is contractible so $$H_2(X)=0$$ and thus $$ri_*$$ factors through zero and thus cannot be the identity on $$\mathbb{Z}=H_2(A)$$ and thus there is no retraction $$r: X \rightarrow A$$

• What you are asking for is clear from the tags, but it wouldn't hurt to include an actual question in the post. – qualcuno Jun 7 '19 at 6:55

Spot on. More generally, if $$X$$ is contractible and $$A \subset X$$ is not acyclic then $$A$$ cannot be a retract of $$X$$. The argument is the same as yours: pick $$q \in \mathbb{N}$$ such that $$0 \neq H_q(A)$$. If we were to have a retraction $$r : X \to A$$, then
$$1_{H_q(A)} = H_q(1_A) = H_q(ri) = H_q(r)H_q(i) \tag{1}$$
Since $$X$$ is contractible, $$H_qr$$ is the zero map (as it has domain zero) and this together with $$(1)$$ implies a contradiction: since $$H_q(A) \neq 0$$, it can't be that $$1_{H_n(A)} = 0$$. Hence no such retraction exists.
• What does the notation $H_q(r)$ mean exactly?? The homomorphism induced by $r$, from $(H_q(X)) \rightarrow H_q(A)?$ – HaKuNa MaTaTa Jun 8 '19 at 22:28
• Precisely! Recall that homology at degree $q$ can be seen as a functor: to each space $X$ we assign an abelian group $H_q(X)$, and to each continuous function $f : X \to Y$ we assign a group morphism $H_qf : H_q(X) \to H_q(Y)$. Sometimes we note $f_*$ in the case of homology or homotopy groups, but the former notation is standard for any functor. – qualcuno Jun 8 '19 at 22:52