# Berger sphere theorem

(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?

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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.

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When $n=2$ or $3$ this is a classic problem at qualifying exams for graduate students. –  YangMills Aug 29 '12 at 17:25