Rational analogs of projective planes

TL;DR

This paper investigates the existence of high-dimensional manifolds with rational homotopy types similar to projective planes, using rational surgery and Diophantine equations to analyze Pontryagin numbers.

Contribution

It introduces a method to determine the existence of such manifolds by translating the problem into solving Diophantine equations related to Pontryagin numbers.

Findings

Reduction of the problem to Diophantine equations

Conditions for Pontryagin numbers satisfying signature and congruences

Framework for identifying manifolds with specified rational homotopy types

Abstract

In this paper, we study the existence of high-dimensional, closed, smooth manifolds whose rational homotopy type resembles that of a projective plane. Applying rational surgery, the problem can be reduced to finding possible Pontryagin numbers satisfying the Hirzebruch signature formula and a set of congruence relations, which turns out to be equivalent to finding solutions to a system of Diophantine equations.