Morse Theory on Graphs

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 which assigns to each oriented edge, e, of Γ, a one-dimensional representation of G (or, alternatively, a weight, αe, in the weight lattice of G. For the assignment, e → αe, to be a schematic description of a " G-action " , these weights have to satisfy certain compatibility conditions: the GKM axioms). As in [GKM] we attach to (Γ, α) an equivariant cohomology ring, H G (Γ) = H(Γ, α). By definition this ring contains the equi-variant cohomology ring of a point, S(g *) = H G (pt), as a subring, and in this paper we will use graphical versions of standard Morse theoretical techniques to analyze the structure of H G (Γ) as an S(g *)-module.