# On dimensions supporting a rational projective plane

## TL;DR

This paper simplifies the classification conditions for rational projective planes, confirms their existence in two new dimensions, and establishes non-existence in others using number theory and surgery techniques.

## Contribution

It introduces a simplified quadratic residue criterion for existence, extends known existence results, and provides new non-existence proofs for rational projective planes.

## Key findings

- Confirmed existence of $ ext{QP}^2$ in two new dimensions.
- Proved non-existence of $ ext{QP}^2$ in certain other dimensions.
- Resolved the existence question for the Spin case.

## Abstract

A rational projective plane ($\mathbb{QP}^2$) is a simply connected, smooth, closed manifold $M$ such that $H^*(M;\mathbb{Q}) \cong \mathbb{Q}[\alpha]/\langle \alpha^3 \rangle$. 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.