We consider effective actions of a compact torus T n−1 on an evendimensional smooth manifold M 2n with isolated fixed points. We prove that under certain conditions on weights of tangent representations, the orbit space is a manifold with corners. Given that the action is Hamiltonian, the orbit space is homeomorphic to S n+1 \ (U 1 . . . U l ) where S n+1 is the (n + 1)-sphere and U 1 , . . . , U l are open domains. We apply the results to regular Hessenberg varieties and manifolds of isospectral Hermitian matrices of staircase form.
Let the compact torus T n´1 act on a smooth compact manifold X 2n effectively with nonempty finite set of fixed points. We pose the question: what can be said about the orbit space X 2n {T n´1 if the action is cohomologically equivariantly formal (which essentially means that H odd pX 2n ; Zq " 0). It happens that homology of the orbit space can be arbitrary in degrees 3 and higher. For any finite simplicial complex L we construct an equivariantly formal manifold X 2n such that X 2n {T n´1 is homotopy equivalent to Σ 3 L. The constructed manifold X 2n is the total space of the projective line bundle over the permutohedral variety hence the action on X 2n is Hamiltonian and cohomologically equivariantly formal. We introduce the notion of the action in j-general position and prove that, for any simplicial complex M , there exists an equivariantly formal action of complexity one in j-general position such that its orbit space is homotopy equivalent to Σ j`2 M .
In this paper we study general torus actions on manifolds with isolated fixed points from combinatorial point of view. The main object of study is the poset of face submanifolds of such actions. We introduce the notion of a locally geometric poset -the graded poset locally modelled by geometric lattices, and prove that for any torus action, the poset of its faces is locally geometric. Next we discuss the relations between posets of faces and GKM-theory. In particular, we define the face poset of an abstract GKM-graph and show how to reconstruct the face poset of a manifold from its GKM-graph.
In this paper we study effective actions of the compact torus on smooth compact manifolds of even dimension with isolated fixed points. It is proved that under certain conditions on the weight vectors of the tangent representation, the orbit space of such an action is a manifold with corners. In the case of Hamiltonian actions, the orbit space is homotopy equivalent to , the complement to the union of disjoint open subsets of the -sphere. The results obtained are applied to regular Hessenberg varieties and isospectral manifolds of Hermitian matrices of step type. Bibliography: 23 titles.
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