Four-dimensional manifolds with pinched positive sectional curvature

We prove that a four-dimensional compact oriented connected Riemannian manifold whose sectional curvatures all lie in the interval [11+33,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[\frac{1}{1+3\sqrt{3}}, 1]$$\end{document} is necessarily definite. In particular, we improve the pinching constants considered by some preceding works. In addition, we show that a four-dimensional compact oriented Einstein manifold whose sectional curvatures all lie in the interval [110,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[\frac{1}{10}, 1]$$\end{document} is either topologically S4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^4$$\end{document} or homothetically isometric to CP2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {CP}^2,$$\end{document} equipped with its standard Fubini-Study metric.