  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
$H^{odd}(X^{2n};\mathbb{Z})=0$). 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 $X^{2n}$ such that
$X^{2n}/T^{n-1}$ is homotopy equivalent to $\Sigma^3L$. 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 $\Sigma^{j+2}M$.
