On orientability, Poincar\'e duality, and connectivity of GKM graphs

Abstract

We investigate a combinatorial notion of orientability for abstract GKM graphs and its connections to graph cohomology in the sense of Guillemin--Zara. In particular, we prove that orientability of the GKM graph is equivalent to Poincar\'e duality of the rational (non-equivariant) graph cohomology algebra. As an application, we prove that orientable GKM graphs remain connected after removing any single vertex.