An Orlik-Raymond type classification of simply connected six-dimensional torus manifolds with vanishing odd degree cohomology

TL;DR

This paper classifies simply connected 6-dimensional torus manifolds with vanishing odd degree cohomology, establishing a correspondence with 3-valent labelled graphs and identifying their diffeomorphism types.

Contribution

It introduces a classification framework linking these manifolds to torus graphs and characterizes their diffeomorphism types using combinatorial methods.

Findings

Manifolds correspond to 3-valent labelled graphs.

Classified manifolds as spheres, quasitoric manifolds, or sphere bundles.

Established a one-to-one correspondence with torus graphs.

Abstract

The aim of this paper is to classify simply connected 6-dimensional torus manifolds with vanishing odd degree cohomology. It is shown that there is a one-to-one correspondence between equivariant diffeomorphism types of these manifolds and 3-valent labelled graphs, called torus graphs introduced by Maeda-Masuda-Panov. Using this correspondence and combinatorial arguments, we prove that a simply connected 6-dimensional torus manifold with vanishing odd degree cohomology is equivariantly diffeomorphic to the 6-dimensional sphere or an equivariant connected sum of copies of 6-dimensional quasitoric manifolds or 4-dimensional sphere bundles over the 2-dimensional sphere.