On the space of self homotopy equivalences of the projective plane (Q1309203)

Given a connected based CW complex \(X\), let \(G(X)\) and \(G_ 0(X)\) denote the spaces of self homotopy equivalences of \(X\) and of based self- homotopy equivalences of \(X\) respectively. For the projective plane \(p^ 2\) the author establishes a homeomorphism \(G(P^ 2) \cong \text{SO}(3) \times (G_ 0(P^ 2)/O(2))\) by showing that the following principal fibre bundles \[ \begin{aligned} \text{SO}(2) & \to G^ +(S^ 2) \to G_ 0^ +(S^ 2)/\text{SO}(2),\\ \text{SO}(2) & \to \text{map}_ 0(S^ 2,P^ 2;\pi)\to \text{map}_ 0(S^ 2,P^ 2; \pi)/\text{SO}(2),\\ \text{SO}(2) & \to G_ 0^ +(P^ 2) \to G_ 0^ +(P^ 2) /\text{SO}(2),\\ \text{SO}(3) & \to G(P^ 2) \to G_ 0(P^ 2) / \text{O}(2) \cong G_ 0^ +(P^ 2) / \text{SO}(2)\end{aligned} \] have cross-sections using the covering map \(\pi: S^ 2 \to P^ 2\), where \(G^ +\) denotes the path component in \(G\) of the identity map.