A rational projective plane ([Formula: see text]) is a simply connected, smooth, closed manifold [Formula: see text] such that [Formula: see text]. An open problem is to classify the dimensions at which such a manifold exists. The Barge–Sullivan rational surgery realization theorem provides necessary and sufficient conditions that include the Hattori–Stong integrality conditions on the Pontryagin numbers. In this paper, we simplify these conditions and combine them with the signature equation to give a single quadratic residue equation that determines whether a given dimension supports a [Formula: see text]. We then confirm the existence of a [Formula: see text] in two new dimensions and prove several non-existence results using factorization of the numerators of the divided Bernoulli numbers. We also resolve the existence question in the Spin case, and we discuss existence results for the more general class of rational projective spaces.

中文翻译：

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关于支持有理射影平面的维度

有理射影平面（[公式：见文本]）是一个简单连通、光滑、封闭的流形 [公式：见文本]，使得 [公式：见文本]。一个悬而未决的问题是对存在这种流形的维度进行分类。Barge-Sullivan 有理手术实现定理提供了必要和充分条件，包括 Pontryagin 数上的 Hattori-Stong 完整性条件。在本文中，我们简化了这些条件并将它们与签名方程结合，给出了一个二次剩余方程，该方程确定给定维度是否支持 [公式：见文本]。然后，我们在两个新维度上确认 [公式：见正文] 的存在，并通过分解伯努利数的分子来证明几个不存在的结果。