Homogeneous spaces, curvature and cohomology

TL;DR

This paper presents new counterexamples of homogeneous Riemannian manifolds in dimensions 13 and 22, challenging the finiteness of rational homotopy types under Gromov's Betti number theorem assumptions.

Contribution

It introduces the first homogeneous manifold counterexamples with specific cohomology ring structures, expanding understanding of curvature and topology relations.

Findings

Counterexamples in dimensions 13 and 22

Homogeneous manifolds with nonnegative curvature and upper curvature bounds

Differences in cohomology ring structures

Abstract

We give new counterexamples to a question of Karsten Grove, whether there are only finitely many rational homotopy types among simply connected manifolds satisfying the assumptions of Gromov's Betti number theorem. Our counterexamples are homogeneous Riemannian manifolds, in contrast to previous ones. They consist of two families in dimensions 13 and 22. Both families are nonnegatively curved with an additional upper curvature bound and differ already by the ring structure of their cohomology rings with complex coefficients. The 22-dimensional examples also admit almost nonnegative curvature operator with respect to homogeneous metrics.