Exercise 10, § 3

In a projective space $\mathbf{P}_n$, let $H_{n,p}$ be the "quadric" defined by the equation

Show that $H_{n,1}$ and $H_{n,n}$ are homeomorphic to $S_{n-1}$. If $2 \leq p \leq n-1$, $H_{n,p}$ is homeomorphic to the space obtained by identifying every point of the product $S_{p-1} \times S_{n-p}$ (considered as a subspace of $\mathbf{R}^p \times \mathbf{R}^{n-p+1}$) with its diametrically opposite point. Every point of $H_{n,p}$ has an open neighbourhood homeomorphic to $\mathbf{R}^{n-1}$.

Show that $H_{3,2}$ is homeomorphic to $S_1 \times S_1$ [identify $S_1 \times S_1$ with $T \times T$, a pair of diametrically opposite points of $S_1 \times S_1$ being identified with a pair of points $(u, v)$ and $(1/2 + u, 1/2 + v)$ of $T \times T$; then consider the mapping $(u, v) \to (u + v, u - v)$ of $T \times T$ into itself].

II) In a projective space $P_n$, let $C_{n,p}$ be the "quadric cone" defined by the equation

Show that the complement of $\{0\}$ in $C_{n,p}$ is homeomorphic to $\mathbf{R} \times H_{n-1,p}$, with the notation of Exercise 10.