Finite topologies for finite geometries

TL;DR

This paper develops a finite topological framework for finite geometries, defining homeomorphisms and exploring invariants like Lefschetz formula and Wu characteristic, revealing new insights into their topological nature.

Contribution

It introduces finite topologies for finite geometries, defines homeomorphisms, and analyzes invariants like Wu characteristic within this finite topological context.

Findings

Lefschetz formula applies to all continuous maps on finite spaces

Wu characteristic and cohomology are topological but not homotopy invariants

Energy theorems relate topological invariants to local interaction energies

Abstract

Without leaving finite mathematics and using finite topological spaces only, we give a definition of homeomorphisms of finite abstract simplicial complexes or finite graphs. Besides exploring the definition in various contexts, we add some remarks like that the general Lefschetz formula works for any continuous map on any finite topological space. We also noted that any higher order Wu characteristic as well as their cohomology are topological invariants which are not homotopy invariants. Energy theorems allow to express these topological invariants in terms of interaction energies of local open sets.