Scalar curvature rigidity of the four-dimensional sphere

Let (M, g) be a closed connected oriented (possibly non-spin) smooth four-dimensional manifold with scalar curvature bounded below by n (n-1). In this paper, we prove that if f is a smooth map of non-zero degree from (M, g) to the unit four-sphere, then f is an isometry. Following ideas of Gromov, we use -bubbles and a version with coefficients of the rigidity of the three-sphere to rule out the case of strict inequality. Our proof of rigidity is based on the harmonic map heat flow coupled with the Ricci flow.