[Paper Review] A primer on computational group homology and cohomology

David Joyner|ArXiv.org|Jun 4, 2007

Homotopy and Cohomology in Algebraic Topology13 references3 citations

TL;DR

This paper provides a computational introduction to group homology and cohomology using the open-source software GAP and SageMath, targeting readers with basic graduate algebra knowledge but no prior cohomology background. It presents foundational theory via projective resolutions and derived functors, demonstrates computations for finite groups, and illustrates key concepts like restriction, inflation, and corestriction through explicit code examples and theoretical results such as the Sylow restriction isomorphism.

ABSTRACT

These are expanded lecture notes of a series of expository talks surveying basic aspects of group cohomology and homology. They were written for someone who has had a first course in graduate algebra but no background in cohomology. You should know the definition of a (left) module over a (non-commutative) ring, what $\zzz[G]$ is (where $G$ is a group written multiplicatively and $\zzz$ denotes the integers), and some ring theory and group theory. However, an attempt has been made to (a) keep the presentation as simple as possible, (b) either provide an explicit reference of proof of everything. Several computer algebra packages are used to illustrate the computations, though for various reasons we have focused on the free, open source packages such as GAP and SAGE.

Motivation & Objective

• To provide a self-contained, accessible introduction to group cohomology and homology for readers with a first-semester graduate algebra background but no prior knowledge of cohomology.
• To bridge abstract homological algebra with concrete computational practice using free, open-source software (GAP and SageMath).
• To demonstrate how to compute cohomology and homology groups for finite groups using explicit algorithms and code.
• To illustrate fundamental functorial properties—restriction, inflation, corestriction—through both theory and computational examples.
• To present key theoretical results, such as the Sylow restriction isomorphism, in a computationally verifiable framework.

Proposed method

• The paper uses projective resolutions and derived functors to define group cohomology as $ H^n(G,A) = \mathrm{Ext}^n_{\mathbb{Z}[G]}(\mathbb{Z}, A) $ and homology as $ H_n(G,A) = \mathrm{Tor}_n^{\mathbb{Z}[G]}(\mathbb{Z}, A) $.
• It introduces the bar resolution as a standard resolution for computing cohomology and homology groups.
• The paper employs the HAP package (via GAP and Sage) to perform actual computations, including chain maps and homology group calculations.
• It demonstrates the use of group homomorphisms to induce maps on cohomology and homology, such as restriction and inflation.
• Theoretical results are verified computationally, e.g., computing $ H_3(S_5, \mathbb{Z}) $ and showing the image of $ H_3(f) $ is the Sylow 2-subgroup.
• The framework is extended to include functorial properties like transitivity of restriction and the inflation-restriction sequence.

Experimental results

Research questions

• RQ1 How can group cohomology and homology be computed algorithmically for finite groups using open-source software?
• RQ2 What is the computational relationship between cohomology groups of a group and its Sylow subgroups?
• RQ3 How do restriction and inflation maps manifest in explicit computations using GAP and Sage?
• RQ4 To what extent can the bar resolution be used to compute cohomology and homology groups in practice?
• RQ5 How do functorial properties like corestriction and transfer behave in concrete computational examples?

Key findings

• The restriction map $ \mathrm{Res}: H^n(G,A) \to H^n(G_p,A) $ restricted to the $ p $-primary component is injective, confirming the Sylow restriction isomorphism.
• For the inclusion $ f: P \hookrightarrow S_5 $, the induced map $ H_3(f): H_3(P,\mathbb{Z}) \to H_3(S_5,\mathbb{Z}) $ has image with Abelian invariants [2,4], matching the 2-primary part of $ H_3(S_5,\mathbb{Z}) \cong \mathbb{Z}_2 \times \mathbb{Z}_4 \times \mathbb{Z}_3 $.
• The inflation map $ \mathrm{Inf}: H^n(G/H, A^H) \to H^n(G,A) $ is well-defined and compatible with the restriction map, forming the inflation-restriction sequence.
• The cohomology and homology groups computed via HAP in GAP and Sage are consistent with theoretical expectations, validating the computational approach.
• The use of the bar resolution and chain maps allows for effective computation of homology and cohomology groups for finite groups.
• The framework enables verification of abstract results, such as the transitivity of restriction and the behavior of induced maps under group homomorphisms.