From Introduction to Topological Manifolds by Lee:
We know $\mathbb S^{n-1}$ is simply connected for $n\ge 3$.
We know that since $\mathbb S^{n-1}$ is a strong deformation retract of $\mathbb R^n\setminus \{0\}$, then $r \circ \iota = \mathrm{Id}_{\mathbb S^{n-1}}$ and $\iota \circ r \simeq \mathrm{Id}_{\mathbb R^n\setminus \{0\}}$, where $\iota:\mathbb S^{n-1} \hookrightarrow \mathbb R^n\setminus \{0\}$ is inclusion.
We do not know homotopy invariance of $\pi_1$.
So, how does the corollary follow from the proposition?
Best Answer
Wow. This is just an oversight. Corollary 7.38 should have been delayed until after the statement of Theorem 7.40, which says that $\pi_1$ is homotopy invariant. I've added a correction.
Thanks for pointing this out.