Torus manifolds with non-abelian symmetries

Abstract

Abstract. Let G be a connected compact non-abelian Lie group and T be 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 = ∅. We show that if there is a torus manifold M with G-action, then the action of a finite covering group of G factors throughwhich has the same orbits as theG-action.We define invariants of torus manifolds with G-action which determine theirG -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 aG-equivariant diffeomorphism. Furthermore, we prove that all admissible 5-tuples may be realised by torus manifolds with G -action, whereG is a finite covering group ofG .

Non-abelian symmetries of quasitoric manifolds

“…Proof. This follows from the results of Section 2 of [5] and the description of the W (G)-action on F given in Theorem 3.1.…”

“…The following structure results were shown in Section 2 of [5]. The group G has a covering group of the form…”

“…Now assume that Mi∈F α M i is non-empty. Then let M be the blow-up of M along Mi∈F α M i (see Section 4 of [5] for details). If we write the characteristic matrix of M in the form…”

“…By the first case there is a G-action on M which extends the torus action. Since the exceptional submanifold is fixed by φ(S 1 ), we can G-equivariantly blow down M along the exceptional manifold to get a G-action on M (see [5,Section 4] for details). Proof.…”

“…In Section 3 we construct the group G from Theorem 1.1. In Section 4 we review the classification of quasitoric manifolds with G-action up to G-equivariant diffeomorphism given in [5]. Moreover, we give a classification of these manifolds up to G-equivariant homeomorphism.…”