Exercise 5, § 4

General Topology, VIII

The sphere $S_5$ being considered as a subspace of $\mathbf{C}^3$ , consider the mapping of $S_5$ into $\mathbf{R}^7$ which sends each point $x = (x_1, x_2, x_3)$ of $S_5$ ( $x_1, x_2, x_3$ complex) to the point $y = (y_i)_{1 \leq i \leq 7}$ of $\mathbf{R}^7$ , where

y_1 = |x_1|^2 - |x_2|^2, \quad y_2 = \mathscr{R}(x_1 \overline{x}_2), \quad y_3 = \mathscr{J}(x_1 \overline{x}_2), \quad y_4 = \mathscr{R}(x_1 \overline{x}_3), y_5 = \mathscr{J}(x_1 \overline{x}_3), \quad y_6 = \mathscr{R}(x_2 \overline{x}_3), \quad y_7 = \mathscr{J}(x_2 \overline{x}_3).

This function has the same value at two points

x, x'

S_5

such that

x' = \zeta x

, where

|\zeta| = 1

. Show that, by passing to the quotient, it induces a homeomorphism of

P_2(\mathbf{C})

onto a subspace of

\mathbf{R}^7

Solution

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