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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 $r: X \rightarrow A$

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

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  • $\begingroup$ Even more generally, a retract of a contractible space must be contractible. $\endgroup$ – Connor Malin Jun 7 '19 at 8:02
  • $\begingroup$ What does the notation $H_q(r)$ mean exactly?? The homomorphism induced by $r$, from $(H_q(X)) \rightarrow H_q(A)?$ $\endgroup$ – HaKuNa MaTaTa Jun 8 '19 at 22:28
  • $\begingroup$ 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. $\endgroup$ – qualcuno Jun 8 '19 at 22:52

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