[Paper Review] On Graph Cohomology and Betti Numbers of Hamiltonian GKM Manifolds

Shisen Luo|arXiv (Cornell University)|Jun 26, 2012

Advanced Combinatorial Mathematics4 references3 citations

TL;DR

This paper introduces characteristic numbers as a combinatorial invariant for GKM graphs of Hamiltonian torus actions on symplectic manifolds, proving that for 8- and 10-dimensional compact Hamiltonian GKM manifolds with moment maps in general position, the even Betti numbers are non-decreasing up to half-dimension. The key contribution is an upper bound on the second Betti number via graph-theoretic connectivity and freeness properties of graph cohomology, establishing a combinatorial counterpart to geometric Betti numbers.

ABSTRACT

In this paper we introduce the concept of characteristic number that are proven to be useful in the study of the combinatorics of graph cohomology. We claim that it is a good combinatorial counterpart for geometric Betti numbers. We then use this concept and tools built along the way to study Hamiltonian GKM manifolds whose moment maps are in general position. We prove some connectivity properties of the their GKM graphs and show an upper bound of their second Betti numbers, which allows us to conclude that these manifolds, in the case of dimension 8 and 10, have non-decreasing even Betti numbers up to half dimension.

Motivation & Objective

• To develop a combinatorial invariant—characteristic numbers—that mirrors geometric Betti numbers in Hamiltonian GKM manifolds.
• To analyze the connectivity and structural properties of GKM graphs under the condition that the moment map is in general position.
• To derive an upper bound on the second Betti number of compact Hamiltonian GKM manifolds using graph-theoretic tools.
• To prove that in dimension 8 and 10, the even Betti numbers of such manifolds are non-decreasing up to half-dimension.
• To establish a correspondence between geometric invariants (Betti numbers) and combinatorial invariants (characteristic numbers) in the context of GKM theory.

Proposed method

• Introduces characteristic numbers $ c_i({\Gamma}) $ as the count of homogeneous generators of degree $ i $ in the equivariant graph cohomology $ H^*_{{\mathbb{T}}^2}({\Gamma}) $, which is shown to be independent of the choice of generators.
• Uses the GKM theorem to realize equivariant cohomology as a subring of tuples of polynomials satisfying edge divisibility conditions defined by the axial function $ \alpha $.
• Applies the concept of $ k $-trimmed subgraphs and the Deleting Lemma to control the rank of graph cohomology in various degrees.
• Establishes that for a graph $ \Gamma $ of type $ A_d $, the number of degree $ d-3 $ generators satisfies $ s_{d-3}(\Gamma) \leq \frac{n_d(\Gamma)}{d-1} + \pi_0(\Gamma) $, where $ n_d $ counts degree-$ d $ vertices and $ \pi_0 $ counts components.
• Uses the freeness of $ H^*_{{\mathbb{T}}^2}({\Gamma}) $ over $ \mathbb{C}[x,y] $ to relate the Betti number $ c_{d-1}(\Gamma) $ to the structure of the graph and its subgraphs.
• Applies connectivity and edge-connectivity conditions to derive the upper bound $ c_{d-1}(\Gamma) \leq \frac{m-2}{d-1} $ for $ c_d(\Gamma) = 1 $, leading to the main result on Betti number monotonicity.

Experimental results

Research questions

• RQ1 Can characteristic numbers serve as a robust combinatorial substitute for geometric Betti numbers in Hamiltonian GKM manifolds?
• RQ2 What topological constraints do the connectivity and edge-connectivity of GKM graphs impose on the Betti numbers of the underlying manifold?
• RQ3 How does the general position of the moment map influence the structure of the GKM graph and its cohomological invariants?
• RQ4 What upper bounds can be established for the second Betti number of compact Hamiltonian GKM manifolds using graph-theoretic methods?
• RQ5 Do the even Betti numbers of 8- and 10-dimensional Hamiltonian GKM manifolds exhibit non-decreasing behavior up to half-dimension?

Key findings

• The characteristic number $ c_i({\Gamma}) $ is well-defined and independent of the choice of homogeneous generators for the free $ \mathbb{C}[x,y] $-module $ H^*_{{\mathbb{T}}^2}({\Gamma}) $, providing a combinatorial analog to geometric Betti numbers.
• For a GKM graph $ \Gamma $ of type $ A_d $, the number of degree $ d-3 $ generators satisfies $ s_{d-3}(\Gamma) \leq \frac{n_d(\Gamma)}{d-1} + \pi_0(\Gamma) $, a key structural bound.
• When $ c_d(\Gamma) = 1 $, the second Betti number satisfies $ c_{d-1}(\Gamma) \leq \frac{m-2}{d-1} $, where $ m $ is the number of vertices.
• For 8- and 10-dimensional compact Hamiltonian GKM manifolds with moment maps in general position, the even Betti numbers are non-decreasing up to half-dimension.
• The GKM graph of such manifolds is $ d $-edge-connected when $ c_d(\Gamma) = 1 $, and the removal of any single edge preserves $ (d-2) $-edge-connectivity.
• The freeness of $ H^*_{{\mathbb{T}}^2}({\Gamma}) $ over $ \mathbb{C}[x,y] $ ensures that the characteristic numbers $ c_i({\Gamma}) $ fully capture the Betti number structure of the underlying manifold.