$X$ is a topological space which has a deformation retraction to $A\subset X$, $a$ is an arbitrary element in $A$ and $i$ is an inclusion function.
A subspace $A$ of $X$ is called a deformation retract of $X$ if there is a homotopy $F:X\times I\rightarrow X$ (called a retract) such that for all $x\in X$ and $a\in A$,
$1.\; F(x,0)=x$,
$2.\; F(x,1) \in A$, and
$3.\; F(a,1)=a$.
We knew $i_{*}$ is one to one. because from (1) we have
$F\circ i$= $1_{A}$
and then
$(F\circ i)_{*}= F_{*}\circ i_{*}= 1_{\Pi_{1}(A,a)} $
Thus $i_{*}$ is injective.
help me to show $i_{*}$ is surjective.
Thanks in advance.