Morse theory on graphs

Abstract

Let $\Gamma$ be a finite d-valent graph and G an n-dimensional torus. An ``action'' of G on $\Gamma$ is defined by a map, $\alpha$, which assigns to each oriented edge e of $\Gamma$ a one-dimensional representation of G (or, alternatively, a weight, ${\alpha }_e$, in the weight lattice of G). For the assignment, $e\to {\alpha }_e$, to be a schematic description of a ``G-action'', these weights have to satisfy certain compatibility conditions: the GKM axioms. We attach to $(\Gamma ,\alpha )$ an equivariant cohomology ring, $H_G(\Gamma )=H(\Gamma ,\alpha )$. By definition this ring contains the equivariant cohomology ring of a point, $\SS(\fg^*) = H_G(pt)$, as a subring, and in this paper we will use graphical versions of standard Morse theoretical techniques to analyze the structure of $H_G(\Gamma )$ as an $\SS(\fg^*)$-module.

23 pages, 1 figure