Rational Ellipticity in Cohomogeneity Two

TL;DR

This paper investigates the rational ellipticity of compact, simply-connected manifolds with isometric Lie group actions of cohomogeneity two, proving ellipticity under the condition of nonnegative sectional curvature.

Contribution

It extends known results by establishing rational ellipticity for cohomogeneity two manifolds with nonnegative curvature, a case not covered by previous theorems.

Findings

Manifolds with cohomogeneity two and nonnegative curvature are rationally elliptic.

The result generalizes the understanding of ellipticity beyond cohomogeneity zero and one cases.

Provides new insights into the topology of manifolds with symmetry and curvature constraints.

Abstract

Let M be a compact, connected and simply-connected Riemannian manifold, and suppose that G is a compact, connected Lie group acting on M by isometries. The dimension of the space of orbits is called the cohomogeneity of the action. If the direct sum of the higher homotopy groups of M, tensored with the field of rational numbers, is a finite-dimensional vector space over the rationals, then M is said to be rationally elliptic. It is known that M is rationally elliptic if it supports an action of cohomogeneity zero or one. When the cohomogeneity is two, this general result is no longer true. However, we prove that M is rationally elliptic in the two-dimensional case under the added assumption that M has nonnegative sectional curvature.