homotopical equivalence of projective real space less a line

Question

let $r$ a projective line of the projective real space. How can i prove that $\mathbb{P} ^3(\mathbb{R}) - r$ is homotopical equivalent to $S^1$?

Answer 1

Think of your projective real 3-space as the group $SO(3)$. The latter fibers over the 2-sphere, with fiber a circle. The fiber can be thought of as a real projective line. Therefore removing a real projective line broduces a circle bundle over the complement of a point on the 2-sphere, namely over a disk. But every circle bundle over a disk is homotopy equivalent to a circle (the bundle is necessarily the trivial bundle).