On Riemannian manifolds with \(\epsilon\)-maximal diameter and bounded curvature (Q1204277)

We study closed Riemannian \(n\)-manifolds \(M\) with \(\varepsilon\)-maximal diameter, i.e. \(\text{Ric}(M)\geq n-1\) and \(\pi- \text{diam}(M)\leq\varepsilon\). We show that there exist two positive numbers \(\varepsilon=\varepsilon(n)\) and \(C(n)\), such that if a closed \(n\)-manifold \(M\) with sectional curvature \(K_ M\geq -K\), \(K>0\), has \(\varepsilon K^{-n/2}\)-maximal diameter, then the total Betti number of \(M\), \(\sum^ n_{i=0} b_ i(M)\), is bounded by \(C(n)\).