Positive Pinching, Volume and Second Betti Number

Abstract.

Our main theorem asserts that for all odd $n \ge 3$ and 0 < $\delta \le 1$, there exists a small constant, $i(n,\delta) > 0$, such that if a simply connected n-manifold, M, with vanishing second Betti number admits a metric of sectional curvature, $\delta \le K_M \le 1$, then the injectivity radius of M is greater than $i(n,\delta)$.