Question

(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?

Answer 1

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=Z. Of course the higher homology groups vanish. Thus the manifold is a simply connected homology sphere, hence by Hurewicz' converse a homotopy sphere.