Structural Conservation (SC) and the Underlying Unification of Algebraic Topology: Paradigm Reconstruction of Isomorphism, Homotopy and Homology

Author: Zhang Suhang

Affiliation: Luoyang School of Mathematics

Date: October 2026

Abstract

Modern algebraic topology and category theory characterise the invariance of mathematical structures under diverse transformations through three core concepts: Isomorphism, Homotopy and Homology. Nevertheless, within traditional mathematical systems, these three notions usually belong to separate categories such as algebra, geometry and homological algebra, leaving room for investigation into their intrinsic unifying logic. This paper introduces the principle of Structural Conservation (SC) to offer a unified macro‑ontological interpretation for these three fundamental concepts. The SC principle states that under arbitrary mappings, deformations or compressions, the apparent scales of a system may undergo drastic changes, whereas its underlying algebraic and topological structures remain relatively conserved. From this viewpoint, isomorphism can be regarded as the “absolute‑conservation layer” of SC under algebraic mappings; homotopy corresponds to the “topological steady‑state layer” of SC under continuous deformations (scale oscillations); homology represents the “invariant‑projection layer” of SC under algebraic‑group quantification. This framework furnishes a novel macro‑perspective for understanding the interconnections across different categories in algebraic topology.

Keywords: Structural Conservation (SC); Isomorphism; Homotopy; Homology; Algebraic Topology; Macro‑Ontology

1. Introduction

The core mission of algebraic topology is to identify invariants of topological spaces under continuous transformations. Isomorphism characterises the absolute equivalence of algebraic structures; homotopy permits spaces to retain topological properties under continuous deformation; homology quantifies the “holes” within spaces by means of algebraic groups. These three powerful tools are often treated as independent mathematical subjects in conventional discussions.

Stepping beyond the limitations of traditional micro‑level definitions, this paper introduces the Structural Conservation (SC) principle to construct a unified geometric‑ontological account. The SC principle states: scales vary, yet structures persist. This paper explores whether isomorphism, homotopy and homology can be understood as manifestations of SC across distinct mathematical categories and transformation scales.

2. Core Axiom of Structural Conservation (SC)

In the limiting evolution, scale compression and iterative transformation of any closed mathematical system, the apparent scale quantities (spatial volume, Hausdorff measure, magnitude of discrete values) may fluctuate violently. By contrast, the essential structural quantities (geometric direction bases, algebraic‑topological architecture, mapping rules) are relatively conserved, precluding structural loss and topological variation.

In short: Scales may vary, structures remain constant; deformation does not break the underlying structure.

3. Isomorphism: The “Absolute‑Conservation Layer” of SC

In category theory, an isomorphism means that two objects preserve all operational structures under a given mapping.

‑ SC perspective: This corresponds to the zero‑deformation state of SC. Regardless of changes in symbols, coordinate systems or carriers (scale variations), the underlying algebraic architecture is conserved. Isomorphism constitutes the most direct and fundamental expression of structural conservation in mathematics: no information or structural collapse occurs.

4. Homotopy: The “Continuous‑Deformation Layer” of SC

Homotopy allows one space to be deformed continuously into another (e.g., a coffee mug reshaped into a doughnut), provided that no tearing or gluing takes place. Homotopy equivalence yields a more macro‑scale perspective of structural conservation than homeomorphism.

‑ SC perspective: This is a typical embodiment of SC. Volume, curvature and scales change drastically under continuous oscillations, while topological invariants (fundamental group, number of holes) stay unchanged. Homotopy may be seen as a macro‑steady‑state realisation of “scales vary, structures remain constant” under continuous geometric evolution.

5. Homology: The “Algebraic‑Invariant Layer” of SC

Homology translates complicated geometric spaces into algebraic groups (e.g., Betti numbers) to count holes. No matter how much geometry is distorted, homology groups remain invariant.

‑ SC perspective: This acts as an algebraic fingerprint of SC under severe geometric compression and high‑dimensional folding. When spatial scales change dramatically, the system preserves homology groups as algebraic invariants. Homology is the quantified projection of SC within the algebraic category.

6. Conclusion

Isomorphism (algebra), homotopy (geometric deformation) and homology (topological quantification) are unified under three hierarchical levels of the Structural Conservation (SC) principle.

Traditional mathematics treats them separately owing to confinement to local micro‑definitions, whereas SC provides macro‑ontological constraints. These three “‑morphism” tools can all be viewed as mathematical forms that emerge when a system maintains its architectural integrity amid drastic scale transformations.

This research program supplies a unified underlying logical perspective for algebraic topology, and puts forward a direction for mathematical inquiry that shifts from local computation to macro‑structural constraints.

Note: This is an original foundational research program. SC is introduced here as a heuristic postulate to provide macro‑ontological explanations. It does not re‑derive the conventional rigorous mathematical definitions of isomorphism, homotopy and homology. Detailed formalisation and proof of the SC axiom will be addressed in subsequent work.