Non-negatively curved GKM orbifolds

TL;DR

This paper investigates non-negatively curved GKM orbifolds and manifolds, showing their cohomology rings match those of specific model orbifolds, and provides a simplified proof for characterizing products of simplices among orbit spaces.

Contribution

It demonstrates the isomorphism of cohomology rings to model orbifolds and simplifies the proof of a key characterization of orbit space products.

Findings

Rational cohomology rings are isomorphic to those of certain model orbifolds.

Models are quotients of isometric actions of finite groups on non-negatively curved torus orbifolds.

Provided a simplified proof of the characterization of products of simplices among orbit spaces.

Abstract

In this paper we study non-negatively curved and rationally elliptic GKM$_4$ manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are quotients of isometric actions of finite groups on non-negatively curved torus orbifolds. Moreover, we give a simplified proof of a characterisation of products of simplices among orbit spaces of locally standard torus manifolds. This characterisation was originally proved in [Wiemeler, Torus manifolds and non-negative curvature, arXiv:1401.0403] and was used there to obtain a classification of non-negatively curved torus manifolds.