Torus actions of complexity one and their local properties

TL;DR

This paper studies torus actions of complexity one on smooth manifolds, establishing conditions under which the orbit space is a topological manifold and introducing the concept of a sponge to describe orbit stratification.

Contribution

It introduces the notion of a sponge to describe orbit stratification and provides conditions for the orbit space to be a topological manifold in complexity one torus actions.

Findings

Orbit space is a closed topological manifold under certain conditions.

The subset of orbits with lower dimension has a specific topology called a sponge.

Manifolds can often be reconstructed from the orbit manifold, sponge, and tangent weights.

Abstract

We consider an effective action of a compact (n-1)-torus on a smooth 2n-manifold with isolated fixed points. We prove that under certain conditions the orbit space is a closed topological manifold. In particular, this holds for certain torus actions with disconnected stabilizers. There is a filtration of the orbit manifold by orbit dimensions. The subset of orbits of dimensions less than n-1 has a specific topology which we axiomatize in the notion of a sponge. In many cases the original manifold can be recovered from its orbit manifold, the sponge, and the weights of tangent representations at fixed points.