TORUS ACTIONS OF COMPLEXITY ONE IN NON-GENERAL POSITION

Let the compact torus T^ act on a smooth compact manifold X^ effectively with nonempty finite set of fixed points. We pose the question: what can be said about the orbit space X^/T^ if the action is cohomologically equivariantly formal (which essentially means that H^(X^;Z)=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^ such that X^2n>/T^ is homotopy equivalent to Σ^3L. The constructed manifold X^ is the total space of a projective line bundle over the permutohedral variety hence the action on X^ is Hamiltonian and cohomologically equivariantly formal. We introduce the notion of an action in j-general poition 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^M