Abstract : In the 60s, Leon Green showed that any compact n-manifold whose scalar curvature is no less than n(n-1) has injectivity radius at most ?, moreover equality is achieved only by the radius 1 sphere. Here we will show that more stringent bounds on the injectivity radius can be shown if on assume some a priori topological information. For instance we will prove that any complete 3-manifold with scalar curvature bigger than 6 has injectivity radius at most 2?/3 except if its fundamental group has odd order. We will also mention some higher dimensional analogues. The proof uses a Gromov's Bonnet-Myers diameter bound for positive scalar curvature metrics on products of 2-spheres with tori.