Orbit spaces of equivariantly formal torus actions of complexity one

TL;DR

This paper investigates the topology of orbit spaces resulting from torus actions of complexity one on manifolds, establishing conditions for homology spheres, homeomorphisms to spheres, and formulas for Betti numbers based on combinatorial data.

Contribution

It introduces the notion of $j$-generality of tangent weights and links it to the acyclicity of orbit spaces, providing new criteria for equivariant formality and topological descriptions.

Findings

Orbit space $Q$ is a homology $(n+1)$-sphere under certain conditions.

If $\pi_1(X)=0$, then $Q$ is homeomorphic to $S^{n+1}$.

Betti numbers are expressed via combinatorial structures in the orbit space.

Abstract

Let a compact torus $T=T^{n-1}$ act on an orientable smooth compact manifold $X=X^{2n}$ effectively, with nonempty finite set of fixed points, and suppose that stabilizers of all points are connected. If $H^{odd}(X)=0$ 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 $(n+1)$-sphere. If, in addition, $\pi_1(X)=0$, 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 $H^{odd}(X)=0$, we prove that $j$-generality of weights implies $(j+1)$-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.