On dimensions supporting a rational projective plane.

A rational projective plane ( ℚ ℙ 2 ) is a simply connected, smooth, closed manifold M such that H ∗ (M ; ℚ) ≅ ℚ [ α ] / 〈 α 3 〉. 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 ℚ ℙ 2 . We then confirm the existence of a ℚ ℙ 2 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. [ABSTRACT FROM AUTHOR]