How is a graph not like a manifold?

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.

23 pages, 3 figures. In v2 we changed the second part of the paper (on the relation between complexity 1 and 0). To clarify the arguments we sacrificed generality: now in Theorem 2 we require GKM graph to bipartite

References in corpus (2)

• How is a graph like a manifold?
• Matroids in toric topology

Cited by in corpus (1)

• Cluster-permutohedra and submanifolds of flag varieties with torus actions