Let a compact torus $T$ of dimension $k$ act effectively on a smooth closed manifold $X$ of dimension $2n$, and assume that the action has nonempty finite set of fixed points. The difference $n-k$ is called the complexity of the action. The action is equivariantly formal if odd-degree cohomology modules of $X$ vanish.

When the action has complexity 0, the orbit space is typically a manifold with corners. Masuda and Panov proved in 2006 that the action of complexity 0 on $X$ is equivariantly formal if and only if all faces of its orbit space are acyclic. In this case, equivariant and ordinary cohomology of $X$ can be described using face rings of simplicial posets. This result generalizes the result of Davis and Januszkiewicz on quasitoric manifolds.

Our general goal is to study actions of complexity one, in particular, equivariantly formal actions. We say that the action of complexity one is in general position, if any $n-1$ of $n$ tangent weights at each fixed point are linearly independent. In the joint work with Masuda we prove that the orbit space of an equivariantly formal action in general position is a homology sphere. This result recovers several previously known particular results if one applies Poincare conjecture in topological category. Moreover, there is a criterion for equivariant formality of the action in terms of its orbit space' structure.

On the other hand, dropping the assumption of general position of tangent weights, the orbit space of equivariantly formal action is not always a sphere. In the joint work with Cherepanov we proved that a triple suspension over any finite simplicial complex can be realized, up to homotopy, as the orbit space of an equivariantly formal (even Hamiltonian) action of complexity one.