On Liftings of Local Torus Actions to Fiber Bundles

TL;DR

This paper introduces a method to lift local torus actions to principal torus bundles, identifying an obstruction class in the first cohomology that determines liftability.

Contribution

It defines the concept of lifting local torus actions to fiber bundles and characterizes the obstruction in cohomological terms.

Findings

Obstruction class resides in the first cohomology group.

Lifting exists if and only if the obstruction class vanishes.

Provides a cohomological criterion for liftability.

Abstract

In this note we define a lifting of a local torus action modeled on the standard representation (we call it a local torus action for simplicity) to a principal torus bundle, and show that there is an obstruction class for the existence of liftings in the first cohomology of the fundamental group of the orbit space with coefficients in a certain module.