the induced homomorphism between fundemental groups Suppose $X,Y$ are topological spaces, $i:X\to Y$ is the inclusion map,$i$ induces a homomorphism $i_{*}:\pi_1(X,x_0)\to \pi_1(Y,x_0)$ such that $i_{*}([f])=[i\circ f]$.
According to the definition ,we have $i_{*}([f])=[i\circ f].$ Since $i$ is the inclusion map,we have $i\circ f(t)=f(t)$ for all $t\in [0,1]$,then we have $i_{*}([f])=[i\circ f]=[f]$.We can conclude that $i_{*}$ is injective without the condition that $Y$ retracts to $X$ .
Can anyone points out the wrong place.Thanks.
 A: The point is that, even if two classes $[f]$ and $[g]$ are different in $\pi_1(X,x_0)$ (that is, $f$ and $g$ are not homotopic in $X$), it can be the case that $[f]=[g]$ in $\pi_1(Y,x_0)$ (when $f$ and $g$ are homotopic in $Y$).
Indeed, when you write "$i_*([f])=[f]$" (which is true!), the "$[f]$" on the left side is different from the "$[f]$" on the right side, even if you write them with the same symbol: the one on the left side is the class of the loops (on $x_0$) homotopic (relatively to $x_0$) to $f$ in $X$, while the one on the right side is the class of the loops homotopic to $f$ in $Y$.
For instance, consider the inclusion $i$ of $D\setminus \{p\}$ into $D$, being $D$ a disk and $p$ one of its points. Let $f$ be a loop on a fixed $x_0 \in D\setminus \{p\}$ such that $f$ goes around $p$. Then clearly $[f]\neq[\sigma_{x_0}]$ in $\pi_1(D\setminus \{p\})$, but of course $[f]=[\sigma_{x_0}]$ in $\pi_1(D)$. Here $\sigma_{x_0}$ denotes the constant loop on $x_0$. Hence in this case $i_*$ is not injective. And indeed $D\setminus \{p\}$ is not a retract of $D$!
Edit. I rewrite everything with the clarifying notation.
The point is that, even if two classes $[f]_X$ and $[g]_X$ are different in $\pi_1(X,x_0)$ (that is, $f$ and $g$ are not homotopic in $X$), it can be the case that $[f]_Y=[g]_Y$ in $\pi_1(Y,x_0)$ (when $f$ and $g$ are homotopic in $Y$).
Indeed, when you write "$i_*([f]_X)=[f]_Y$" (which is true!), the "$[f]_X$" on the left side is different from the "$[f]_Y$" on the right side: $[f]_X$ the one on the left side is the class of the loops (on $x_0$) homotopic (relatively to $x_0$) to $f$ in $X$, while $[f]_Y$ is the class of the loops homotopic to $f$ in $Y$.
For instance, consider the inclusion $i$ of $D\setminus \{p\}$ into $D$, being $D$ a disk and $p$ one of its points. Let $f$ be a loop on a fixed $x_0 \in D\setminus \{p\}$ such that $f$ goes around $p$. Then clearly $[f]_{D\setminus\{p\}}\neq[\sigma_{x_0}]_{D\setminus\{p\}}$ in $\pi_1(D\setminus \{p\})$, but of course $[f]_D=[\sigma_{x_0}]_D$ in $\pi_1(D)$. Here $\sigma_{x_0}$ denotes the constant loop on $x_0$. Hence in this case $i_*$ is not injective. And indeed $D\setminus \{p\}$ is not a retract of $D$!
