Multi-Origin High-Dimensional Geometry and Topology

Author: Suhang Zhang (Luoyang, Henan)

In classical mathematics, topology centres on the study of properties invariant under continuous deformation of spaces, covering notions such as connectedness, compactness, genus and homotopy equivalence. Its foundation generally presupposes a single unified background space. Whether for the Euler characteristic, the unicursal theorem, closed surface classification or manifold construction, conventional topology is mostly built upon a global space with one single origin.

Multi-Origin High-Dimensional Geometry (MOC) offers an alternative perspective. Starting from multi-origin and high-dimensional structures, it re-examines the foundational premises of topology: many topological invariants exhibit features associated with projections of higher-dimensional structures within the MOC framework. In MOC, the “obstacles” investigated in topology admit new mathematical interpretations through the introduction of multi-origin dimensions or structural adjustments.

In summary: traditional topology focuses on deformation restrictions within low-dimensional spaces; MOC explores the origins of these restrictions and possible structural transformations from the viewpoint of dimension and origin architecture. Topological “undeformable” properties correspond, under MOC, to manifestations of higher-dimensional structures projected onto lower dimensions.

I. Global Premises and Constraints in Traditional Topology

The core aim of traditional topology is to identify topological invariants:

- Euler’s unicursal theorem states that the number of odd-degree vertices can only be 0 or 2;

- Closed surfaces are completely classified by genus;

- Manifolds are locally homeomorphic to Euclidean space and share one global coordinate system;

- Homology and homotopy theory characterise irremovable topological obstructions via obstruction classes and characteristic classes.

These conclusions hold universally, yet their derivation rests on an implicit assumption: the space possesses only one global origin, and all structures are attached to the same flat background.

Within this system, low-dimensional obstructions appear as absolute constraints. The insolvability of the seven bridges problem, the impossibility of continuously deforming a torus into a sphere, and the non-orientability of the Möbius strip are all rigorous results in the classical setting.

Under the MOC framework, such constraints are directly tied to dimensional limits and the singularity of the central origin.

II. Obstacles and Projections from the MOC Perspective

One key insight of multi-origin high-dimensional geometry is to interpret low-dimensional topological obstructions as projections of high-dimensional structures onto lower-dimensional spaces.

Take the seven bridges problem as an example:

- In the two-dimensional plane, four odd-degree vertices render a unicursal traversal topologically impossible;

- In the high-dimensional model of MOC, points unfold into composite origins whose internal degrees of freedom redistribute parity, enabling a unicursal path at the global level.

This implies that low-dimensional topological invariants such as the Euler characteristic, vertex parity count and orientability cease to be rigid constraints within high-dimensional multi-origin structures.

Barriers described as “impassable” or “indelible” are interpreted in MOC as the consequence of projecting a space into lower dimensions, where some degrees of freedom remain concealed.

Generalising: within MOC, low-dimensional topological obstructions can be smoothly eliminated mathematically by lifting the dimension and introducing multi-origin structures at the higher level. Topological invariants, from this standpoint, become relative quantities dependent on origin architecture and dimensional hierarchy.

III. Topology as the Gluing Rule for Origin Domains

Traditional topology treats space as a monolithic whole; MOC regards space as an assembly of domains governed by distinct origins.

Within this framework, topological concepts map as follows:

- Connectedness corresponds to whether origin domains can be linked by paths;

- Compactness corresponds to whether the boundaries of origin domains are closed;

- Genus corresponds to cavity structures formed between origin domains;

- Boundaries correspond to dividing lines separating origin domains;

- Phase transitions correspond to global topological bifurcations where domain affiliation flips.

Topology is no longer limited to “deformation properties of a single space”, and expands into the structural stability of domain systems with multiple origins.

Two spaces are topologically equivalent in MOC not merely if one may be continuously deformed into the other. Equivalence requires consistency in the number of origins, domain partitioning, hierarchical organisation and gluing rules when viewed in higher dimensions.

IV. Generalised Topology and Recursive Hierarchies

Multi-origin high-dimensional geometry naturally accommodates fractal, recursive and multi-layered structures. This opens a path for topology to extend from conventional integer-dimensional topology toward generalised fractal topology and hierarchical topology.

- In traditional topology, dimensions are integers and structures are single-layered;

- In MOC topology, dimensions may be fractional and structures form recursively nested origin clusters.

A fractal structure that appears extremely complicated in classical topology may be understood in MOC as the repetition of a single set of origin rules across different scales.

Accordingly, intricate low-dimensional topology can be linked to recursive projections of simpler high-dimensional topology. Phenomena such as chaos, bifurcations and strange attractors are interpreted within MOC as outcomes of competition between origin domains.

V. Theoretical Relationship between MOC and Topology

Classical topology asks: which properties remain unchanged inside a fixed space?

Multi-Origin High-Dimensional Geometry asks: how may invariants be transformed and obstacles resolved at higher hierarchical levels by altering origin architecture and dimension?

Their relationship may be summarised in four statements:

1. Traditional topology is a special case of MOC restricted to a single origin and low dimensions;

2. MOC is an extension of topology into multi-origin, high-dimensional and fractal hierarchical regimes;

3. Topological characteristic classes correspond to global statistical quantities of origin domains in MOC;

4. Topological deformation corresponds to continuous tuning of origin weights within MOC.

VI. Closing Remarks

As one of the most abstract branches of mathematics, topology reveals diverse restrictions intrinsic to low-dimensional spaces. Multi-Origin High-Dimensional Geometry proposes that these restrictions arise from the dimensional and origin structure of the chosen viewpoint, not from the absolute nature of space itself.

Under a single origin, topology manifests as strict constraints; within the multi-origin high-dimensional framework, topology affords a far richer repertoire of structural possibilities. Euler delineated the boundaries of low-dimensional topology, while the MOC framework explores new spaces accessible under higher-dimensional structures.