On the topology of G-manifolds with finitely many non-principal orbits

TL;DR

This paper explores the topology of compact G-manifolds with finitely many non-principal orbits, characterizing their orbit spaces, cohomogeneity, and homotopy types, and providing new examples and classifications.

Contribution

It classifies possible orbit spaces for such manifolds, links singular orbits to odd cohomogeneity, and constructs new examples with specific orbit configurations.

Findings

Non-principal orbits imply odd cohomogeneity.

Classified possible orbit spaces for manifolds with finitely many non-principal orbits.

Constructed infinite families of examples with one or two singular orbits.

Abstract

We study the topology of compact manifolds with a Lie group action for which there are only finitely many non-principal orbits, and describe the possible orbit spaces which can occur. If some non-principal orbit is singular, we show that the Lie group action must have odd cohomogeneity. We pay special attention to manifolds with one and two singular orbits, and construct some infinite families of examples. To illustrate the diversity within some of these families, we also investigate homotopy types.