Retractions and Isomorphisms of Fundamental Groups

Question

Suppose there is a retraction from $$S^1 \times D^2 \to S^1 \times S^1.$$ Does that then induce an isomorphism $$\pi_1(S^1) \times \pi_1(D^2) \cong \pi_1(S^1) \times \pi_1(S^1)?$$ Which is obviously not true, otherwise $$\mathbb{Z} \cong \mathbb{Z}^2.$$ This yields the result I want, however, I'm not sure if there is such an induced isomorphism between the fundamental groups.

Answer 1

You should write one more arrow.

A retraction $\rho$ is a continuous map $\rho : S^1\times D^2 \to S^1\times S^1$ such that if $i$ is the natural embedding $S^1\times S^1 \to S^1\times D^2$ then $\rho \circ i = Id$. From this we get $\rho _*\circ i_*= Id$. But $i_*$ is not injective, as it kills the loop wich is the boundary of the disc, contradiction.