I want to use homology to solve the following problem: Prove that the circle represented by the blue arc in the picture is not a retract of the Klein bottle. (See the attached picture of the Klein bottle)
I know the $q$-th singular homology module of the Klein bottle $K$, and some about singular homology, for example relative homology, MV-sequence, and so on. At first I tried to use the following: If $A$ is a retract of $X$ then $H_q(X) = H_q(A)\oplus H_q(X,A)$. But it does not seem to be helpful for this problem. Do I miss something simple?
Thank you for help!
I think it's because $H_1(A)=\Bbb Z$ is generated by $[a]$, while $H_1(K)$ is $\Bbb Z_2\oplus \Bbb Z$, where $\Bbb Z_2$ is generated by $[a]$ and $\Bbb Z$ is generated by $[b]$. So if $A$ were a retract of $K$, then we would have an injection $$H_1(A)=\langle[a]\rangle\hookrightarrow H_1(K)=⟨[a],[b]⟩$$ sending $[a]$ to $[a]$, which is impossible since $[2a]=0$ in $H_1(K)$.