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Finite topologies for finite geometries

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arxiv 2301.03156 v1 pith:DJX2VMKE submitted 2023-01-09 math.CO

Oliver Knill This is my paper · ORCID
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keywords finitetopologicalinvariantsdefinitionabstractallowbesidescharacteristic
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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.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Dehn Sommerville Manifolds

    math.CO 2025-08 reject novelty 6.0 of 10

    Dehn-Sommerville manifolds form a broad class of finite simplicial complexes that the paper claims to endow with Dehn-Sommerville face symmetries, level-set closure, chromatic bound 2q+2, and monoid closure under joins.

  2. Euler Characteristics of Random Manifolds

    math.CO 2026-07 conditional novelty 5.0 of 10

    For a random codimension-1 level set H in a simplicial complex G, E[χ(H)] equals 2−2K(G)−χ(G), with K the curvature functional built from the f-vector.

  3. Interacting Geodesics on Discrete Manifolds

    math.DS 2025-05 conditional novelty 5.0 of 10

    A reversible, deterministic evolution of signed particles on the frame bundle of a finite simplicial complex is defined, generalizing single-particle geodesic flow to interacting multi-particle systems.

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