Scalar curvature rigidity of the four-dimensional sphere

Let ( M , g ) be a four-dimensional closed connected oriented (possibly non-spin) Riemannian manifold whose scalar curvature is bounded below by 12. We prove that, if f is a smooth distance non-increasing map of non-zero degree from ( M , g ) to the unit four-sphere, then f is an isometry. This removes the spin condition in Llarull’s scalar curvature rigidity theorem for spheres in dimension four.