The complex projective plane, CP^2, is a 4-dimensional manifold, defined as the set of nonzero vectors in C^3, modulo the equivalence relation z ~ cz for all nonzero complex numbers c. The action of complex conjugation descends (from C^3) to CP^2, and though this action is not free, the quotient space is still a manifold. I will describe a proof via Lie group actions that this quotient space is actually homeomorphic to the 4-sphere.