Caputi - On finite generation in magnitude (co)homology, and its torsion

Abstract

The aim of the talk is to show that magnitude cohomology yields finitely generated modules over categories of (directed) graphs with bounded genus. We will use the theory of quasi-Groebner categories, as recently developed by Sam and Snowden, and present the case of directed graphs. Then, we will discuss two main consequences. First, the ranks of magnitude homology of graphs with bounded genus, in a fixed degree, grow at most polynomially in the number of vertices. Second, the order of its torsion, in each fixed degree, is bounded.

Recap

Setting: $G$ is an undirected, connected, non-simple graph (quivers) Regular morphisms

Defintion A minor morphism $\varphi :G^{\prime }\to G$ is a map, that allows edges and vertices to be deleted. (As long as it contracts the graph)

Recall: $M{C}_{\ast ,\ast }(G)={⨁}_{l\ge 0}{M}_{\ast ,\ast }(G)$ where $M_{k,l}(G)$ are the Chains of the Magntde homology.

We now define the Rank: $M{H}_{k,l}:CGraph\to Mo{d}_{\mathbb{R}}$. Computations:

• $G=K_n$ complete graph, then the The Magnitude homology is equal to the homology chain.s
• $E(G)=\varnothing$: The $0,0$ Magnitude is generated freely by the set of vertices.
• $G$ is tree: The $l,l$ Homology is generated by the Edges for all $l>0$

Question: Is there Torion in MH? Yes, sometimes.

Representation theory of sets

Application