Homotopy Self-Equivalences of $\mathbb{R}{P}^2\times S^2$ and a Braid for non-orientable $4$-Manifolds

We compute the group of based homotopy classes of self-homotopy equivalences of the non-orientable $4$-manifold ${\text{RP}}^2\times S^2$. It is an elementary abelian $2$-group of rank $5$. The subgroup of classes whose based lift to the universal cover $S^2\times S^2$ has degree $+1$ has rank $4$ and is isomorphic to the group of free homotopy classes. The group maps onto the group of self-equivalences of the algebraic $2$-type, which has order $8$, with kernel ${\pi }_4({\text{RP}}^2\times S^2)\cong (\mathbb{Z}\mathrm{/}2{)}^2$ generated by pinch-fold maps; that these maps are independent is shown by an adjunction argument. We then construct a non-orientable analogue of the Hambleton–Kreck braid of exact sequences in dimension $4$, for non-orientable manifolds with spin universal cover, stated for finite fundamental group. We apply the braid to ${\text{RP}}^2\times S^2$ and to ${\text{RP}}^4$, and for the Hambleton–Hillman manifold with fundamental group $(\mathbb{Z}\mathrm{/}2{)}^2$ we determine the normal $1$-type and show that self-equivalences act on ${\pi }_2$ through diagonal sign matrices. For ${\text{RP}}^2\times S^2$ the pinch-fold maps are detected by twisted spin bordism, and, assuming facts from surgery theory stated as hypotheses, exactly one of the three non-trivial pinch-fold maps is homotopic to a homeomorphism.