skills.rest
bneb
bnebCOMMUNITY
Explore all skills by this author →
GitHub

homological_cohomological_arguments

Compute homology and cohomology invariants using Lean 4 templates and mathlib4.

1|Updated Mar 14, 2026
Search Tags:#lean4#cohomology#homology#algebraic-topology#exact-sequences#mayer-vietoris#cup-product
One-click install
npx skills add https://github.com/bneb/perqed --skill homological-cohomological-arguments
Or copy as Structured Prompt for Agent▼
Please help me install this Agent Skill.
Skill: homological_cohomological_arguments
Source: https://github.com/bneb/perqed/tree/main/.agents/skills/homological_cohomological_arguments
Command: npx skills add https://github.com/bneb/perqed --skill homological-cohomological-arguments

SYSTEM DOCUMENTATION & REQUIREMENTS

What problem does it solve?

Compute homology and cohomology invariants to obstruct or establish the existence of maps, sections, and global solutions in topology and algebraic settings.

Core Features & Use Cases

  • Compute homology groups H_n(X) and cohomology rings H^*(X) for given spaces.
  • Use exact sequences (long exact sequence, Mayer-Vietoris) and cup products to detect obstructions to existence of maps or retractions.
  • Apply to topology problems like fixed-point theory, non-contractibility, and homotopy equivalence tests.

Quick Start

Input a topological space description and request computation of its homology groups and obstruction results.

Frequently Asked Questions about homological_cohomological_arguments

High-intent search queries and answers about installing and using this skill.

FAQPage Schema
How do I compute homology groups to detect obstructions to maps in algebraic topology?▼

Exact sequences detect obstructions by linking homology groups across spaces. This Skill supports applying long exact sequences and Mayer-Vietoris sequences to identify when maps, sections, retracts, or global solutions cannot exist in topology problems.

Can I use cup products to prove non-contractibility or test homotopy equivalence?▼

Yes, you can use cup products to prove non-contractibility and test homotopy equivalence. The Skill supports computing cup-product obstructions within cohomology rings to detect topological properties like fixed-point theory and non-contractibility.

Does this approach work with Lean 4 and mathlib4 for computing cohomology invariants?▼

To compute cohomology invariants, provide a topological space description as input. The Skill then computes the requested homology groups H_n and cohomology rings H^n, applying exact sequences and cup products to deliver obstruction results.

What are the limitations of using exact sequences for detecting obstructions to sections and retracts?▼

Mayer-Vietoris sequences compute homology groups by decomposing a topological space into simpler overlapping subspaces. This Skill applies Mayer-Vietoris sequences alongside long exact sequences and cup products to detect obstructions to maps and retractions.

Related Skills

Explore More Education & Research Skills→
fmhallfmhall
1

lean-array-list

Provide Lean 4 proof strategies and lemmas for ByteArray, Array, and List operations.

Community
Advanced
r-irber-irbe
2

math-topology-analysis

Structure reasoning workflows for topology, functional analysis, and dynamical systems.

Community
Intermediate
bnebbneb
1

generating_functions

Derive recurrences and closed forms for sequences using formal power series in Lean 4.

Community
Advanced
GuitarAlchemistGuitarAlchemist

ix-ktheory

Compute algebraic K-theory invariants K0 and K1 for graph structures.

Official
Advanced