关于支持有理射影平面的维数
On dimensions supporting a rational projective plane
摘要：
有理射影平面（$\mathbb{QP}^2$）是一个单连通的光滑闭流形$M$，使得$H^*(M;\mathbb{Q}) \cong \mathbb{Q}[α]/\langle α^3 \rangle$。一个开放的问题是分类的尺寸，在这种流形存在。Barge-Sullivan有理手术实现定理给出了包含Pontryagin数的Hattori-Stong完整性条件的充要条件.在本文中，我们简化了这些条件，并将它们与签名方程结合起来，得到一个二次剩余方程，用于确定给定的维数是否支持$\mathbb{QP}^2$。然后，我们确认存在一个$\mathbb{QP}^2$在两个新的层面，并证明了几个不存在的结果，使用因式分解的分子划分伯努利数。我们还解决了存在性问题的自旋的情况下，我们讨论的存在性结果更一般的类的合理射影空间。
英文摘要：
A rational projective plane ($\mathbb{QP}^2$) is a simply connected, smooth, closed manifold $M$ such that $H^*(M;\mathbb{Q}) \cong \mathbb{Q}[α]/\langle α^3 \rangle$. 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 article, 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 $\mathbb{QP}^2$. We then confirm existence of a $\mathbb{QP}^2$ in two new dimensions and prove several non-existence results using factorizations of numerators of 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.
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