The existence of generating families for the cohomology ring of a graph

Abstract

Let Γ be a finite d-valent graph and G an n-dimensional torus. An action of G on Γ is defined by a map that assigns to each oriented edge e of Γ a 1-dimensional representation of G (or, alternatively, a weight α e in the weight lattice of G. For the assignment e→ α e to be a schematic description of a G-action, these weights have to satisfy certain compatibility conditions). We attach to ( Γ, α) an equivariant cohomology ring, H( Γ, α). By definition, this ring contains the equivariant cohomology ring of a point, H G(pt)= S( g ∗) , as a subring, and in this paper we use graphical versions of standard Morse theoretical techniques to analyze the structure of H( Γ, α) as an S( g ∗) -module.