Here $S^3$ is the $3$-sphere, and the Hopf link is described as in here.
In my class, we have done work to show that $S^3 \setminus (\text{Hopf link}) \cong \mathbb{R} \times T^2$, where $T^2$ is the usual torus. For the purpose of this question I will just treat this as given. The note goes on to say that $\mathbb{R} \times T^2$ is homotopically equivalent to $T^2$, hence we can make the inference that $\pi_1(S^3 \setminus (\text{Hopf link})) \cong \pi_1(T^2) \cong \mathbb{Z}^2$.
I have two questions about this last claim.
I want to make sure I understand the homotopy equivalence mentioned. If we see $\mathbb{R} \times T^2$ as living in $\mathbb{R}^5$ (i.e. consisting of $5$-tuple points), can we see the retraction as resulted from “removing” the first coordinates?
I see how we can have $\pi_1(S^3 \setminus (\text{Hopf link})) \cong \pi_1(\mathbb{R} \times T^2)$, as fundamental groups of homeomorphic spaces are isomorphic. But what about the second inference? Are fundamental groups of homotopically equivalent spaces also isomorphic?