Classification of locally standard torus actions

TL;DR

This paper classifies locally standard torus actions on manifolds by describing their structure through combinatorial and cohomological data, providing a comprehensive framework for understanding these symmetries.

Contribution

It introduces a classification scheme for locally standard torus actions using triples of combinatorial and cohomological invariants, advancing the understanding of such actions.

Findings

Classification of actions via triples (Q, lambda, c)

Description of quotient manifolds with corners and labellings

Cohomological encoding of twisting in the actions

Abstract

An action of a torus T on a manifold M is locally standard if, at each point, the stabilizer is a sub-torus and the non-zero isotropy weights are a basis to its weight lattice. The quotient M/T is then a manifold-with-corners, decorated by a so-called unimodular labelling, which keeps track of the isotropy representations in M, and by a degree two cohomology class with coefficients in the integral lattice of the Lie algebra of T, which encodes the "twistedness" of M over M/T. We classify locally standard smooth actions of T, up to equivariant diffeomorphisms, in terms of triples (Q,lambda,c), where Q is a manifold-with-corners, lambda is a unimodular labelling, and c is a degree two cohomology class with coefficients in the integral lattice.