A torus manifold M is a 2n-dimensional orientable manifold with an effective action of an ndimensional torus such that M T = ∅. In this paper, we discuss the classification of torus manifolds which admit an invariant metric of non-negative curvature. If M is a simply connected torus manifold which admits such a metric, then M is diffeomorphic to a quotient of a free linear torus action on a product of spheres. We also classify rationally elliptic torus manifolds M with H odd (M ; Z) = 0 up to homeomorphism.
Abstract. In this work, it is shown that a simply-connected, rationallyelliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an application, simply-connected, rationally-elliptic manifolds admitting slice-maximal torus actions are classified up to equivariant rational homotopy. The case where the rational-ellipticity hypothesis is replaced by non-negative curvature is also discussed, and the Bott Conjecture in the presence of a slice-maximal torus action is proved.
Abstract. In 2006 Masuda and Suh asked if two compact non-singular toric varieties having isomorphic cohomology rings are homeomorphic. In the first part of this paper we discuss this question for topological generalizations of toric varieties, so-called torus manifolds. For example we show that there are homotopy equivalent torus manifolds which are not homeomorphic. Moreover, we characterize those groups which appear as the fundamental groups of locally standard torus manifolds.In the second part we give a classification of quasitoric manifolds and certain six-dimensional torus manifolds up to equivariant diffeomorphism.In the third part we enumerate the number of conjugacy classes of tori in the diffeomorphism group of torus manifolds. For torus manifolds of dimension greater than six there are always infinitely many conjugacy classes. We give examples which show that this does not hold for six-dimensional torus manifolds.
Abstract. Let T be a torus of dimension ≥ k and M a T -manifold. M is a GKM k -manifold if the action is equivariantly formal, has only isolated fixed points, and any k weights of the isotropy representation in the fixed points are linearly independent.In this paper we compute the cohomology rings with real and integer coefficients of GKM 3 -and GKM 4 -manifolds which admit invariant metrics of positive sectional curvature.
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 .
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