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.