Let $A$ be a topological subspace of $X$. Then we have exact sequences of homotopy groups for pair spaces: $$ \ldots \rightarrow \pi_n(A,x_0) \xrightarrow{i_{*}} \pi_n(X,x_0) \xrightarrow{j_{*}} \pi_n(X,A;x_0) \xrightarrow{\partial} \pi_{n-1}(A,x_0) \rightarrow \ldots, $$ Assume that there exist a homotopy $f_t:X \mapsto X$ such that $f_0 = \text{id},\ f_1(X) \subset A$. Then $j_*$ is the trivial homomorphism.
Any idea how to prove it?
Thanks.
Edit:
So, this is wrong statement. In this picture you can see a counterexample.
$f_t$ is retraction into circle around the hole. As you can see $a \not \in \text{Ker}\,j_{*}$.
So, maybe $A$ should be path-connected?