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In particular, we introduce a conformal invariant $\\beta(M^4,[g]) \\geq 0$ defined on conformal four-manifolds satisfying a `positivity' condition; it follows from \\cite{CGY03} that if $0 \\leq \\beta(M^4,[g]) < 4$, then $M^4$ is diffeomorphic to $S^4$. Our main result of this paper is a `gap' result showing that if $b_2^{+}(M^4) > 0$ and $4 \\leq \\beta(M^4,[g]) < 4(1 + \\epsilon)$ for $\\epsilon > 0$ small enough, then $M^4$ is diffeomorphic to $\\mathbb{CP}^2$. 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Chang","submitted_at":"2018-09-16T17:28:00Z","abstract_excerpt":"We extend the sphere theorem of \\cite{CGY03} to give a conformally invariant characterization of $(\\mathbb{CP}^2, g_{FS})$. In particular, we introduce a conformal invariant $\\beta(M^4,[g]) \\geq 0$ defined on conformal four-manifolds satisfying a `positivity' condition; it follows from \\cite{CGY03} that if $0 \\leq \\beta(M^4,[g]) < 4$, then $M^4$ is diffeomorphic to $S^4$. Our main result of this paper is a `gap' result showing that if $b_2^{+}(M^4) > 0$ and $4 \\leq \\beta(M^4,[g]) < 4(1 + \\epsilon)$ for $\\epsilon > 0$ small enough, then $M^4$ is diffeomorphic to $\\mathbb{CP}^2$. 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