[Paper Review] Finite covers of graphs, their primitive homology, and representation theory

Benson Farb, Sebastian Hensel|arXiv (Cornell University)|Oct 27, 2016

Geometric and Algebraic Topology8 references8 citations

TL;DR

This paper establishes a representation-theoretic framework to study the primitive homology of finite regular graph covers, linking topological properties of covers to the structure of their first homology as representations of the deck group. It proves that primitive homology is constrained by the set of irreducible representations admitting fixed vectors under primitive elements of the fundamental group, and shows that for abelian or 2-step nilpotent groups with rank ≥ 3, primitive homology exhausts the full homology, though counterexamples exist in rank 2.

ABSTRACT

Consider a finite, regular cover $Y o X$ of finite graphs, with associated deck group $G$. We relate the topology of the cover to the structure of $H_1(Y;\mathbb{C})$ as a $G$-representation. A central object in this study is the {\em primitive homology} group $H_1^{\mathrm{prim}}(Y;\mathbb{C})\subseteq H_1(Y;\mathbb{C})$, which is the span of homology classes represented by components of lifts of primitive elements of $π_1(X)$. This circle of ideas relates combinatorial group theory, surface topology, and representation theory.

Motivation & Objective

• To understand the structure of the first homology $ H_1(Y;\mathbb{C}) $ of a finite regular cover $ Y \to X $ as a representation of the deck group $ G $.
• To determine when the primitive homology $ H_1^{\mathrm{prim}}(Y;\mathbb{C}) $, spanned by lifts of primitive loops, equals the full homology $ H_1(Y;\mathbb{C}) $.
• To establish a dictionary between topological data of covers and representation-theoretic invariants of the deck group.
• To investigate the role of primitive elements in the fundamental group in generating homology classes under group actions.
• To resolve the question of whether all irreducible representations of the deck group appear in the primitive homology, particularly for abelian and nilpotent groups.

Proposed method

• Uses the Chevalley-Weil formula to express $ H_1(Y;\mathbb{C}) $ as a sum of irreducible representations of the deck group $ G $, with multiplicities determined by the rank of the base graph's fundamental group.
• Defines the primitive homology $ H_1^{\mathrm{prim}}(Y;\mathbb{C}) $ as the $ \mathbb{C} $-span of homology classes lifted from primitive elements in $ \pi_1(X) $, forming a $ G $-subrepresentation.
• Introduces the set $ \mathrm{Irr}^{\mathrm{pr}}(\phi,G) $ of irreducible representations $ V \in \mathrm{Irr}(G) $ for which some primitive element of $ F_n $ acts nontrivially with a fixed vector in $ V $, to characterize which irreps can appear in primitive homology.
• Applies surface topology and covering space theory to derive constraints on the primitive homology, particularly through the use of subsurfaces and boundary curves.
• Employs algorithms based on mapping class group actions to compute which group elements are realized by nonseparating simple closed curves.
• Constructs explicit examples using group presentations to demonstrate cases where $ H_1^{\mathrm{prim}}(Y;\mathbb{C}) \neq H_1(Y;\mathbb{C}) $, especially in rank 2 covers.

Experimental results

Research questions

• RQ1 Under what conditions on the deck group $ G $ and the covering map $ Y \to X $ does the primitive homology $ H_1^{\mathrm{prim}}(Y;\mathbb{C}) $ equal the full homology $ H_1(Y;\mathbb{C}) $?
• RQ2 Which irreducible representations of the deck group $ G $ appear in the primitive homology $ H_1^{\mathrm{prim}}(Y;\mathbb{C}) $?
• RQ3 Can the condition $ \mathrm{Irr}^{\mathrm{pr}}(\phi,G) = \mathrm{Irr}(G) $ be guaranteed for all surjective homomorphisms $ \phi: F_n \to G $, and what does this imply for the homology?
• RQ4 Are there finite covers of rank 2 graphs where primitive homology is strictly smaller than full homology, and if so, what group-theoretic properties cause this?
• RQ5 How can algorithms based on mapping class group actions be used to detect which group elements are realized by simple closed curves in a cover?

Key findings

• The primitive homology $ H_1^{\mathrm{prim}}(Y;\mathbb{C}) $ is contained in the direct sum of irreducible representations $ V_i \in \mathrm{Irr}^{\mathrm{pr}}(\phi,G) $, each with multiplicity $ (n-1)\dim(V_i) $, plus the trivial representation.
• For abelian or 2-step nilpotent deck groups $ G $, and when $ \operatorname{rank}(\pi_1(X)) \geq 3 $, it holds that $ \mathrm{Irr}^{\mathrm{pr}}(\phi,G) = \mathrm{Irr}(G) $, implying $ H_1^{\mathrm{prim}}(Y;\mathbb{C}) = H_1(Y;\mathbb{C}) $ under mild additional conditions on the center of $ G $.
• In rank 2 covers (e.g., wedge of two circles), counterexamples exist where $ H_1^{\mathrm{prim}}(Y;\mathbb{C}) \neq H_1(Y;\mathbb{C}) $, even for 2-step nilpotent groups.
• An explicit example is constructed with a group $ G $ acting freely and linearly on a sphere, where no simple closed curve lifts to a trivial element in homology, but some primitive elements do, showing $ H_1^{\mathrm{scc}}(Y;\mathbb{C}) \neq H_1^{\mathrm{prim}}(Y;\mathbb{C}) $.
• The paper provides an algorithm to compute the image of simple closed curves under a homomorphism $ q: \pi_1(\Sigma) \to G $, enabling detection of which group elements are realized by nonseparating simple curves.
• The exact sequence $ \mathbb{C}[\sum \partial R_j] \to \bigoplus_j \mathbb{C}[\partial R_j] \to \bigoplus_i \mathbb{C}[\hat{\alpha}_i] \to \mathrm{Span}_{H_1(X)}\{\hat{\alpha}_1,\ldots,\hat{\alpha}_n\} \to 0 $ is established, showing that homology relations arise from boundary curves of subsurfaces.