Strengthened injectivity radius bounds for manifolds with positive scalar curvature

Green's inequality shows that a compact Riemannian manifold with scalar curvature at least n (n-1) has injectivity radius at most, and that equality is achieved only for the radius 1 sphere. In this work we show how extra topological assumptions can lead to stronger upper bounds. The topologies we consider are S²^n-k-2ᵏ for n 7 and 0 k 2 and 3-manifolds with positive scalar curvature except lens spaces L (p, q) with p odd. We also prove a strengthened inequality for 3-manifolds with positive scalar curvature and large diameter. Our proof uses previous results of Gromov and Zhu.