Exercise 4, § 3

If $n > 1$, there exists an essential mapping $f$ of $S_1$ into $P_n$ such that $f(S_1) \neq P_n$ [take $f(S_1)$ to be the image under $\varphi$ of a diameter of $B_n$; this image is a projective line]. Deduce that, if $n > 1$, $P_n$ is homeomorphic to neither $S_n$ nor $B_n$ (use Exercise 3 of § 3 and Exercise 3 of § 2).