On the mapping class groups of ${\mathbb{P}}^1$-bundles over ${\mathbb{P}}^2$

In this article we compute the mapping class group of the total space $S(\xi )$ of the sphere bundle of a 3-dimensional real vector bundle $\xi$ over the complex projective plane ${\mathbb{P}}^2$ with $⟨p_1(\xi ),[{\mathbb{P}}^2]⟩=8n+5$. Examples of these manifolds include the Milnor hypersurface $M_1$ and its generalizations $M_k=\{(x,y)\in {\mathbb{P}}^2\times {\mathbb{P}}^2\mathrm{\mid }\sum x_i^k{y}_i=0\}$ with $k$ odd.