(Berger, 1958) Let M be a closed nmanifold with sec ≥ 1 and injp > π/2 for some p ∈ M, then M is (n − 1)connected and hence a homotopy sphere. I don't quite understand the "hence".Must a n1 connected manifold be a homotopy sphere? After we get M is n1 connected,how can we prove M is a homotopy sphere?
By Hurewicz (n1)connected implies vanishing of the first n1 homology groups. Since the manifold is closed and (by simple connectedness) also orientable, we have H_n=Z. Of course the higher homology groups vanish. Thus the manifold is a simply connected homology sphere, hence by Hurewicz' converse a homotopy sphere. 

