Equivariant cohomological rigidity of certain $T$-manifolds

TL;DR

This paper introduces a new class of smooth manifolds with torus actions, called locally $k$-standard $T$-manifolds, and investigates their topological properties and conditions under which their equivariant cohomology uniquely identifies them.

Contribution

It defines the category of locally $k$-standard $T$-manifolds and analyzes their topological invariants and cohomology, extending known classes like toric and quasitoric manifolds.

Findings

Locally $k$-standard $T$-manifolds include toric and quasitoric manifolds.

Equivariant cohomology can distinguish these manifolds up to weakly equivariant homeomorphism.

Topological properties such as fundamental groups are characterized for these manifolds.

Abstract

We introduce the category of {\it locally $k$-standard $T$-manifolds} which includes well-known classes of manifolds such as toric and quasitoric manifolds, good contact toric manifolds and moment-angle manifolds. They are smooth manifolds with well-behaved actions of tori. We study their topological properties, such as fundamental groups and equivariant cohomology algebras. Then, we discuss when the torus equivariant cohomology algebra distinguishes them up to weakly equivariant homeomorphism.