(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 $H_n={\mathbb Z}$. 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.
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$\begingroup$ When $n=2$ or $3$ this is a classic problem at qualifying exams for graduate students. $\endgroup$ Commented Aug 29, 2012 at 17:25