AIM-TOPOLOGY-0102 · Complete counterexample

A Four-Point Counterexample to Excision for Directed Cubical Homology of Closure Spaces

Alper Ferudun

Manuscript 2 September 2026 · Online 2 September 2026

math.ATUnrefereed preprint

Abstract

Bubenik and Milićević defined cubical singular homology theories of Čech closure spaces from an interval and either the product or inductive product. Excision is known for the product theories and was left open for the three inductive theories. We give a four-point counterexample for the directed interval \(J_+\) . For \(X=J_+\mathbin{\boxdot}J_+\) , an interior cover \(\{U,V\}\) has \[ H^{(J_+,\boxdot)}_1(V,U\cap V;\Z)\cong\Z, \] generated by the difference of the two coordinate edges. Under inclusion into \((X,U)\) this class is the boundary of the identity square. Hence the excision map is not injective. The calculation is integral, exact, and accompanied by two independently implemented finite-chain checks. This is an unrefereed note and makes no absolute priority claim.

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Record

Affiliation
Mercury Software GmbH
Result
Complete counterexample
Categories
math.AT
Manuscript
2 September 2026
Online release
2 September 2026
Version
1.0 (typesetting revision 2026-09-05)
License
Creative Commons Attribution 4.0 International

Files and verification

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PDF typesetting revised 5 September 2026: author/contact layout and disclosure placement only. The DOI links to the original archived edition; the mathematical content is unchanged.

Citation

Alper Ferudun, “A Four-Point Counterexample to Excision for Directed Cubical Homology of Closure Spaces,” EulerSolve Research Papers, AIM-TOPOLOGY-0102, 2026. https://doi.org/10.5281/zenodo.22245271.

BibTeX
@misc{Ferudun2026AimTopology0102,
  author = {Ferudun, Alper},
  title = {A Four-Point Counterexample to Excision for Directed Cubical Homology of Closure Spaces},
  year = {2026},
  howpublished = {EulerSolve Research Papers},
  url = {https://eulersolve.org/papers/aim-topology-0102/},
  doi = {10.5281/zenodo.22245271},
  note = {AIM-TOPOLOGY-0102; unrefereed preprint}
}