[Paper Review] Morse theory on graphs

Victor Guillemin, Catalin Zara|ArXiv.org|Jul 26, 2000

Algebraic structures and combinatorial models3 references3 citations

TL;DR

This paper develops a Morse-theoretic framework for analyzing the equivariant cohomology ring $ H_G(\Gamma) $ of a GKM graph $ (\Gamma, \alpha) $, where $ \Gamma $ is a finite, d-valent graph and $ \alpha $ assigns weights to edges. By leveraging polarizing vectors and graphical Morse theory, the authors prove that $ H_G(\Gamma) $ is a free module over $ \SS(\mathfrak{g}^*) $, determine the number of generators in each degree via symmetric polynomials and Vandermonde determinants, and provide a combinatorial description of these generators using symmetric function theory and integral operations on weight data.

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, $α$, which assigns to each oriented edge e of $Γ$ a one-dimensional representation of G (or, alternatively, a weight, $α_e$, in the weight lattice of G). For the assignment, $e o α_e$, to be a schematic description of a ``G-action'', these weights have to satisfy certain compatibility conditions: the GKM axioms. We attach to $(Γ, α)$ an equivariant cohomology ring, $H_G(Γ)=H(Γ,α)$. 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 $H_G(Γ)$ as an $\SS(\fg^*)$-module.

Motivation & Objective

• To understand the structure of the equivariant cohomology ring $ H_G(\Gamma) $ as a module over $ \SS(\mathfrak{g}^*) $, the cohomology of a point.
• To determine whether $ H_G(\Gamma) $ is a free $ \SS(\mathfrak{g}^*) $-module and, if so, to compute the number of generators in each degree.
• To provide a combinatorial description of the generators of $ H_G(\Gamma) $ using symmetric polynomials and weight data on the graph.
• To establish a graphical Morse-theoretic framework for analyzing equivariant cohomology without relying on smooth manifolds.
• To generalize results from GKM theory to abstract graphs equipped with axial functions satisfying the GKM axioms.

Proposed method

• Define a polarizing vector $ \xi \in \mathfrak{g} $ such that $ \alpha_e(\xi) \neq 0 $ for all edges $ e $, enabling an orientation of the graph via $ \alpha_e(\xi) > 0 $.
• Use the orientation induced by $ \xi $ to define a height function on vertices, enabling Morse-theoretic decomposition of the cohomology ring.
• Apply graphical analogues of standard Morse theoretical techniques, such as the GKM condition and the structure of the one-skeleton, to analyze the cohomology ring.
• Utilize symmetric polynomials and the theory of Vandermonde matrices to describe the generators of $ H_G(\Gamma) $ as elements of $ \SS(\mathfrak{g}^*) $-modules.
• Introduce an integral operation $ \int_\Delta $ on maps from a finite set $ \Delta $ to the quotient field of $ \SS(W) $, which characterizes elements of the cohomology space $ H(\Delta, \tau) $.
• Prove that any element $ g \in H(\Delta, \tau) $ can be written as a linear combination $ \sum_{k=0}^{d-1} g_k \tau^k $ with $ g_k \in \SS(W) $, using properties of symmetric polynomials and determinants.

Experimental results

Research questions

• RQ1 Is the equivariant cohomology ring $ H_G(\Gamma) $ a free module over $ \SS(\mathfrak{g}^*) $?
• RQ2 If $ H_G(\Gamma) $ is free, how many generators does it have in each degree?
• RQ3 Can the generators of $ H_G(\Gamma) $ be described in a combinatorial way using symmetric functions and weight data?
• RQ4 How does the graphical Morse-theoretic approach on $ \Gamma $ reflect the structure of the cohomology ring?
• RQ5 What is the role of the axial function $ \alpha $ in determining the module structure of $ H_G(\Gamma) $?

Key findings

• The equivariant cohomology ring $ H_G(\Gamma) $ is a free module over $ \SS(\mathfrak{g}^*) $, as established through Morse-theoretic decomposition and the use of polarizing vectors.
• The number of generators of $ H_G(\Gamma) $ in degree $ m $ is given by $ \sum_{k=0}^{d-1} \lambda_{m-k,n} $, where $ \lambda_{i,n} $ is the number of monomials of degree $ i $ in $ n $ variables.
• Every element $ g \in H(\Delta, \tau) $ can be uniquely expressed as $ \sum_{k=0}^{d-1} g_k \tau^k $ with $ g_k \in \SS(W) $, providing a canonical basis for the cohomology module.
• The inverse of a Vandermonde matrix associated with weight data provides explicit formulas for the coefficients in the basis expansion, with $ (A^{-1})_{1j} = \prod_{k \neq j} \left( \frac{-X_k}{X_j - X_k} \right) $.
• The integral operation $ \int_\Delta g = \sum_{k=1}^d \frac{g(v_k)}{\prod_{j \neq k} (\tau(v_k) - \tau(v_j))} $ characterizes elements of $ H(\Delta, \tau) $, and is used to reconstruct the polynomial coefficients $ g_k $.
• The cohomology space $ H(\Delta, \tau) $ is isomorphic to the space of maps $ g: \Delta \to \SS(W) $ satisfying the integral condition, and this isomorphism is compatible with the grading on $ \SS(W) $.