Characterization of locally standard, $\mathbb{Z}$-equivariantly formal manifolds in general position

TL;DR

This paper characterizes a class of manifolds with specific symmetry properties, showing their cohomology can be described similarly to known torus manifolds, especially in higher dimensions.

Contribution

It provides a new characterization of locally standard, $bZ$-equivariantly formal manifolds in general position and relates their cohomology to that of equivariantly formal torus manifolds.

Findings

For dimensions at least 10, such manifolds correspond to certain torus manifolds.

The cohomology with $bZ$-coefficients matches that of equivariantly formal torus manifolds.

The paper establishes a link between labeled GKM graphs and manifold structures.

Abstract

We give a characterization of locally standard, $\mathbb{Z}$-equivariantly formal manifolds in general position. In particular, we show that for dimension $2n$ at least $10$, to every such manifold with labeled GKM graph $\Gamma$ there is an equivariantly formal torus manifold such that the restriction of the $T^n$-action to a certain $T^{n-1}$-action yields the same labeled graph $\Gamma$, thus showing that the (equivariant) cohomology with $\mathbb{Z}$-coefficients of those manifolds has the same description as that of equivariantly formal torus manifolds.