Orbit spaces of equivariantly formal torus actions

Abstract

Let a compact torus T " T n´1 act on a smooth compact manifold X " X 2n effectively, with nonempty finite set of fixed points, and suppose that stabilizers of all points are connected. If H odd pXq " 0 and the weights of tangent representation at each fixed point are in general position, we prove that the orbit space Q " X{T is a homology pn`1q-sphere. If, in addition, π 1 pXq " 0, then Q is homeomorphic to S n`1 . We introduce the notion of j-generality of tangent weights of torus action. For any action of T k on X 2n with isolated fixed points and H odd pXq " 0, we prove that j-generality of weights implies pj `1q-acyclicity of the orbit space Q. This statement generalizes several known results for actions of complexity zero and one. In complexity one, we give a criterion of equivariant formality in terms of the orbit space. In this case, we give a formula expressing Betti numbers of a manifold in terms of certain combinatorial structure that sits in the orbit space.

How is a graph not like a manifold?

“…vanishes for i ă j and i " j `1. 4) is not stated in [3] explicitly, however, its proof follows the same lines as the proof of item (3). We outline the main ideas of this proof.…”

“…(see details in [3]). The sequence (3.4) is acyclic for equivariantly formal actions according to [10] (for rational coefficients) and Franz-Puppe [15] (over integers presuming connectedness of stabilizers).…”

“…In Section 3 we prove several statements about homological properties of orbit spaces of torus actions. Most of the arguments there follow the lines of [3]. In Section 4 we prove Theorem 1.…”

“…(3) We fix an integer j, and call the action j-independent, if every j tangent weights, at every fixed point x P X T are linearly independent. In the previous works [3,4] we used a different terminology: actions with this property were called actions in j-general position. With these properties satisfied, there holds Theorem 1.…”

“…As an application of Theorem 1, we describe equivariant cohomology algebra for complexity one actions in general position. Acyclicity arguments are applied to extend the study of complexity one actions in general position, started in [3]. An action of T n´1 on X 2n is called an action in general position, if it is pn ´1q-independent.…”