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 π, 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 S²ⁿ⁻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.