Manifolds with non-negative sectional curvature

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1. Diameter Pinching

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Problem 1.1.
Does the Grove-Petersen-Wu Finiteness Theorem hold in dimension $4$ , i.e. are there only finitely many diffeomorphism types in the class of $4$ -dimensional Riemannian manifolds satisfying $\sec_M \geq - \Lambda^2, \ \ \operatorname{diam}(M) \leq D, \ \ \operatorname{Vol}(M) \geq V ?$
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Problem 1.2.
If $\sec_M \geq 1$ and $\operatorname{diam}(M) > \frac{\pi}{2}$ , must $M$ be diffeomorphic to $S^n$ ?
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Problem 1.3.
Let $M^n$ be a manifold for which $\sec_M \geq 1$ and $\operatorname{diam} (M) \geq \frac{\pi}{2} - \varepsilon(n)$ . Is $M$ homeomorphic to a manifold $M'$ , where $\sec_{M'} \geq 1$ , $\operatorname{diam}(M') \geq \frac{\pi}{2}$ ? This problem may be easier to address if we also assume that $\operatorname{Vol}(M) \geq V$ and $\varepsilon = \varepsilon(n, V)$ .

Cite this as: AimPL: Manifolds with non-negative sectional curvature, available at http://aimpl.org/nnsectcurvature.