Elements of finite geometry I

2026-08-02

AI Summary

This study systematically investigates fundamental theoretical problems in finite geometry—encompassing graphs and finite abstract simplicial complexes—with a focus on twelve key topics, including the Gauss–Bonnet theorem, Poincaré–Hopf theorem, Euler formula, Euler–Poincaré relation, fixed-point theorems, and cohomology theory. By extending classical results from differential geometry and algebraic topology to discrete finite structures, the work establishes a unified framework for combinatorial geometry. Integrating methods from combinatorial topology, graph theory, discrete differential geometry, cohomology theory, and index theory, the authors rigorously formulate discrete analogues of several classical theorems, thereby laying a solid theoretical foundation for future research in finite geometry.

Abstract

This is a snapshot of a first part on a possibly much longer text on finite geometries, meaning graphs or finite abstract simplicial complexes. In in this first batch we review 12 subjects: Gauss-Bonnet, Poincare-Hopf, Index expectation, Euler's gem, Euler-Poincare,Unimodularity, Brouwer-Lefschetz, Sphere formula, Level sets, Index formula, Quadratic cohomology and Higher characteristic.

Problem

• finite geometry
• simplicial complexes
• Euler characteristic
• cohomology
• index theory

Innovation

• finite geometry
• simplicial complexes
• discrete topology
• Euler characteristic
• index theory