Torus actions of complexity one in non-general position

TL;DR

This paper studies torus actions of complexity one on manifolds, showing that the orbit space's homology can be arbitrarily complex in higher degrees and constructing specific examples with prescribed homotopy types.

Contribution

It introduces the notion of actions in j-general position and constructs equivariantly formal manifolds with orbit spaces homotopy equivalent to suspensions of arbitrary complexes.

Findings

Homology of orbit spaces can be arbitrary in degrees 3 and higher.

Constructed manifolds are total spaces of projective line bundles over permutohedral varieties.

Existence of actions with orbit spaces homotopy equivalent to suspensions of any simplicial complex.

Abstract

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 $\Sigma^{j+2}M$.