An Orlik–Raymond type classification of simply connected $6$-dimensional torus manifolds with vanishing odd-degree cohomology

Shintarô Kuroki

Vol. 280 (2016), No. 1, 89–114

DOI: 10.2140/pjm.2016.280.89

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 and Panov. Using this correspondence and combinatorial arguments, we prove that a simply connected $6$-dimensional torus manifold with $H^{odd}\left(M\right)=0$ is equivariantly diffeomorphic to the $6$-dimensional sphere $S^6$ or an equivariant connected sum of copies of $6$-dimensional quasitoric manifolds or $S^4$-bundles over $S^2$.

Dedicated to Professor Mikiya Masuda on his 60th birthday.

Keywords

torus manifold, torus graph, GKM graph, equivariant connected sum

Mathematical Subject Classification 2010

Primary: 57S25

Secondary: 94C15

Milestones

Received: 4 September 2014

Accepted: 20 April 2015

Published: 28 December 2015