On the quotient spaces of \(S^2\times S^2\) under the natural action of subgroups of \(D_4\) (Q2765976)

The fourth diedral group \(D_4\) acts naturally on the manifold \(S^2\times S^2\). So, to each of the 10 subgroups of \(D_4\) corresponds a quotient space of \(S^2\times S^2\). Five of such quotient spaces were determined in [\textit{W. S. Massey}, Geom. Dedicata 2, 371-374 (1973; Zbl 0273.57019)] and they are diffeomorphic to \(S^2\times S^2\), \(\mathbb{C}\mathbb{P}^2\), \(\mathbb{R} \mathbb{P}^2\times\mathbb{R} \mathbb{P}^2\), \(\mathbb{R}\mathbb{P}^4\) and \(S^4\). In the paper under review, the author investigates all the 10 quotient spaces and considers all the existing 15 quotient maps. Four of the remaining 5 quotient spaces are proved to be diffeomorphic to \(\overline{\mathbb{C}\mathbb{P}}^2\) (endowed with the orientation opposite to \(\mathbb{C}\mathbb{P}^2)\), \(\mathbb{R}\mathbb{P}^2\times S^2\), \(S^2\times\mathbb{R}\mathbb{P}^2\) and the Grassmannian manifold \(G_2 (\mathbb{R}^4)\) of all unoriented 2-planes through the origin of \(\mathbb{R}^4\). The problem to find a familiar manifold diffeomorphic to the last quotient space, namely, \(S^2\times S^2/\text{span}\left(\begin{smallmatrix} 0 & -1\\ 1 & 0 \end{smallmatrix} \right)\), where \(D_4\) has been identified with the space of orthogonal matrices fixing \((\pm 1,\pm 1)\) in \(\mathbb{R}^2\) remains open.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00014].