On the Rational Hyperbolicity problem

TL;DR

This paper characterizes when certain symmetric manifolds are rationally elliptic based on the flatness of their quotient spaces, resolving several conjectures in the field of rational homotopy theory.

Contribution

It establishes a precise criterion linking rational ellipticity of manifolds with symmetries to the flatness of their quotient spaces, under mild conditions.

Findings

Rationally elliptic manifolds with certain symmetry conditions are characterized by flat quotient spaces.

Counterexamples exist where rational ellipticity coexists with hyperbolic quotient spaces.

The results resolve multiple conjectures in the rational homotopy of symmetric manifolds.

Abstract

We prove that a compact simply connected manifold $M$ with a variationally complete $G$-action satisfying certain mild conditions (e.g. trivial principal isotropy, or simply connected principal orbits) is rationally elliptic if and only if $M/G$ is flat. This answers several conjectures and problems regarding the rational homotopy of manifolds with symmetries. On the other hand, without the extra conditions we find examples of rationally elliptic $G$-manifolds $M$ where $M/G$ admits a hyperbolic metric.