I am solving the following exercise.
Show that there is no deformation retract $r:X\rightarrow A$ where $X=\bar{\Bbb{B}^2}\lor\bar{\Bbb{B}^2}$ and $A=\Bbb{S}^1\lor\Bbb{S}^1$.
As I understand this, if we have a deformation retract, then two spaces are homotopy equivalent and have the same fundamental group right?
But looking at the prove of this exercise given by our TA, I think something went wrong, this is the solution.
He stated that $S^1$ is a retract of $\Bbb{S}^1\lor\Bbb{S}^1$ but if he mans $\Bbb{S}^1$ then I don't think this is true since we don't have the same fundamental group. But another $S^1$ wasn't defined. So I'm a bit confused.
Would be nice if someone can take a look and explain me where the mistake is.
Additional question Is this an iff statement when we say that if X is a deformation retract of Y, and $\Pi(X)=\Pi(Y)$?
Thanks.