Exercise 1, § 3

General Topology, VI

Let $f$ be the mapping of $S_2$ into $\mathbf{R}^4$ such that

f(x_1, x_2, x_3) = (x_1^2 - x_2^2, x_1 x_2, x_1 x_3, x_2 x_3).

This function has the same value at diametrically opposite points of

S_2

. Show that on passing to the quotient it defines a homeomorphism of

P_2

onto a subspace of

\mathbf{R}^4

Show likewise that the mapping $g$ of $S_3$ into $\mathbf{R}^6$ defined by

g(x_1, x_2, x_3, x_4) = (x_1^2 - x_2^2, x_1 x_2, x_1 x_3 + x_2 x_4, x_3^2 - x_4^2, x_3 x_4, x_1 x_4 - x_2 x_3)

defines, on passing to the quotient, a homeomorphism of

P_3

into

\mathbf{R}^6

Solution

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