(Berger, 1958) Let M be a closed n-manifold 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 n-1 connected manifold be a homotopy sphere?
After we get M is n-1 connected,how can we prove M is a homotopy sphere?



Berger sphere theorem
By Hurewicz, (n-1)-connected implies vanishing of the first n-1 homology groups. Since the manifold is closed and (by simple connectedness) also orientable, we have $H\_n={\mathbb Z}$. Of course the higher homology groups vanish. Thus the manifold is a simply connected homology sphere, hence by the converse of Hurewicz a homotopy sphere.


