How is a graph not like a manifold?

TL;DR

This paper explores the topological and algebraic properties of GKM graphs arising from torus actions on manifolds, establishing conditions for their acyclicity and describing their equivariant cohomology in relation to face algebras.

Contribution

It introduces new acyclicity conditions for face posets of torus actions and links equivariant cohomology to face algebras, enhancing understanding of GKM graph structures.

Findings

J-independency implies (j+1)-acyclicity of skeleta

Provides necessary conditions for GKM graphs of manifolds

Describes equivariant cohomology via face algebra of simplicial posets

Abstract

For an equivariantly formal action of a compact torus $T$ on a smooth manifold $X$ with isolated fixed points we investigate the global homological properties of the graded poset $S(X)$ of face submanifolds. We prove that the condition of $j$-independency of tangent weights at each fixed point implies $(j+1)$-acyclicity of the skeleta $S(X)_r$ for $r>j+1$. This result provides a necessary topological condition for a GKM graph to be a GKM graph of some GKM manifold. We use particular acyclicity arguments to describe the equivariant cohomology algebra of an equivariantly formal manifold of dimension $2n$ with an $(n-1)$-independent action of $(n-1)$-dimensional torus, under certain colorability assumptions on its GKM graph. This description relates the equivariant cohomology algebra to the face algebra of a simplicial poset. Such observation underlines certain similarity between actions of complexity one and torus manifolds.