Torus manifolds with non-abelian symmetries

TL;DR

This paper classifies torus manifolds with non-abelian symmetries by introducing invariants called admissible 5-tuples, which determine their equivariant diffeomorphism types, and shows how these manifolds relate to specific Lie group actions.

Contribution

It introduces admissible 5-tuples as invariants for classifying torus manifolds with non-abelian symmetries and establishes their realization by specific group actions.

Findings

Finite covering groups of G factor through a product of classical groups and tori.

Admissible 5-tuples uniquely determine simply connected torus manifolds with G-action.

All admissible 5-tuples can be realized by manifolds with specific group actions.

Abstract

Let (G) be a connected compact non-abelian Lie-group and (T) a maximal torus of (G). A torus manifold with (G)-action is defined to be a smooth connected closed oriented manifold of dimension (2\dim T) with an almost effective action of (G) such that (M^T\neq \emptyset). We show that if there is a torus manifold (M) with (G)-action then the action of a finite covering group of (G) factors through (\tilde{G}=\prod SU(l_i+1)\times\prod SO(2l_i+1)\times \prod SO(2l_i)\times T^{l_0}). The action of (\tilde{G}) on (M) restricts to an action of (\tilde{G}'=\prod SU(l_i+1)\times\prod SO(2l_i+1)\times \prod U(l_i)\times T^{l_0}) which has the same orbits as the (\tilde{G})-action. We define invariants of torus manifolds with (G)-action which determine their (\tilde{G}')-equivariant diffeomorphism type. We call these invariants admissible 5-tuples. A simply connected torus manifold with (G)-action is determined by its admissible 5-tuple up to (\tilde{G})-equivariant diffeomorphism. Furthermore we prove that all admissible 5-tuples may be realised by torus manifolds with (\tilde{G}")-action where (\tilde{G}") is a finite covering group of (\tilde{G}').