On equivariantly formal 2-torus manifolds

TL;DR

This paper characterizes equivariantly formal 2-torus manifolds via local standardness and face acyclicity of the orbit space, linking their topological properties to group actions and providing criteria for the existence of regular involutions.

Contribution

It establishes a precise criterion for equivariant formality of 2-torus manifolds based on orbit space properties, extending the understanding of their topological and group action structure.

Findings

Equivariant formality is characterized by local standardness and face acyclicity.

Provides conditions for 2-torus manifolds to admit regular m-involutions.

Connects topological properties with group action features in 2-torus manifolds.

Abstract

A 2-torus manifold is a closed connected smooth n-manifold with a non-free effective smooth $\mathbb{Z}^n_2$-action. In this paper, we prove that a 2-torus manifold is equivariantly formal if and only if the $\mathbb{Z}^n_2$-action is locally standard and every face of its orbit space (including the whole orbit space) is mod 2 acyclic. Our study is parallel to the study of torus manifolds with vanishing odd-degree cohomology by M. Masuda and T. Panov. As an application, we determine when such kind of 2-torus manifolds can have regular m-involutions (i.e. involutions with only isolated fixed points of the maximum possible number).