DVG Topological Ball: Dynamic Genus Vectors and a Virtual–Real Topological Transformation System

Author: Zhang Suhang

School: Luoyang School of Mathematics

October 2026

Abstract

Classical differential topology is built on the foundation of “topological invariance”: the topological type of a manifold and the number of genera are invariants under homeomorphic deformation, and geometric entities and topological holes are fixed structures with no mechanism for mutual transformation. Traditional topology studies only the static properties of already-formed manifolds and cannot describe the generation, annihilation, or phase-transition evolution of topological structures.

Based on the framework of Dynamic Vector Geometry (DVG), this paper establishes a dynamic topological model of the topological ball. It introduces the genus vector as the fundamental dynamic unit of topological degrees of freedom, breaks the classical presupposition of topological invariants, and constructs a bidirectional evolution system between “topological virtual quantities (genus vectors)” and “geometric real quantities (entity manifold structures).”

Under the governance of the basic DVG coupling equation, genus vectors can condense to generate local geometric entities; conversely, geometric entities can also be deconstructed and retracted, transforming into pure topological genus vectors. This paper completes the paradigm construction of dynamic topology, upgrading topology from a static geometric property to a dynamic object that can evolve, undergo phase transitions, and be mutually transformed. This is an expansion and innovation of the classical topological system.

Keywords: Dynamic Vector Geometry; DVG; topological ball; genus vector; dynamic topology; topological phase transition; virtual–real topological transformation

I. Introduction

Since Poincaré founded modern topology, the core premise of the entire topological system has remained constant:

The topological structure of a manifold is invariant.

The underlying logic of classical topology is:

1. Manifolds pre-exist;

2. Genus, holes, and connectivity are topological invariants;

3. Continuous deformation does not change topological type;

4. Entity geometry and topological holes have no evolution, no birth or annihilation, and no interchange.

This system has supported the development of modern topology, differential geometry, and physical topology, but it has a fundamental limitation:

It can study only “topologies that already exist,” not “how topology comes into being, how it changes, and how it disappears.”

Dynamic Vector Geometry (DVG) completely changes the underlying geometric paradigm: spacetime and geometry are no longer static backgrounds, but dynamic products of the coupled evolution of curvature vector fields and angular momentum vector fields. Based on DVG field dynamics, this paper further extends the geometric system and constructs a topological ball model, achieving for the first time:

· Topology is no longer an invariant property, but a dynamic variable;

· Genus is no longer a static label, but an evolvable vector field;

· Topological virtual structures and geometric real structures undergo bidirectional transformation and mutual generation.

This model does not negate classical topology; rather, it encompasses and extends classical topology: classical static topology is a special case of DVG dynamic topology under the condition of “frozen topological evolution.”

II. Inherent Limitations of Classical Topology

2.1 Classical Definition of Genus

In classical topology, the genus g is the number of holes of a manifold and satisfies topological invariance:

\text{Homeomorphic deformation} \implies \Delta g = 0

All topological analysis is built on the premise of no topological phase transition and no birth or annihilation of structures.

2.2 Three Major Shortcomings of Classical Topology

1. No topological generation mechanism: it cannot explain the origin of holes and topological structures;

2. No topological annihilation mechanism: entity structures cannot degenerate into topological information;

3. Complete separation of real and virtual: geometric entities and topological holes belong to completely different geometric categories, with no transformation channel.

For this reason, classical topology is a static, passive, descriptive geometric tool and does not possess dynamic evolutionary capability.

III. Core Architecture of the DVG Topological Ball

3.1 Core Innovation: Vectorization of Genus

DVG abandons the classical scalar genus and defines the genus vector \mathbf{g}:

· It belongs to the dynamic field quantities of topological degrees of freedom;

· It is uniformly governed by the basic DVG coupling equation;

· It can evolve, superpose, condense, and be deconstructed.

3.2 Basic DVG Governing Equation

All virtual–real transformations, topological evolution, and birth and annihilation of structures in the topological ball obey the fundamental DVG equation:

\nabla_\mu \mathbf{K}^{\mu\nu}=\alpha \mathbf{J}^\nu

The coupled dynamics of the curvature vector field and the angular momentum vector field simultaneously governs:

1. Spacetime geometric evolution (at the GR level)

2. Topological structure phase transitions (at the topological level)

This realizes a unified dynamic source for geometry and topology.

3.3 Bidirectional Transformation Principle of the Topological Ball

This paper establishes two new topological evolution mechanisms:

(1) Virtual → Real: Genus Vectors Condense into Geometric Entities

Pure topological information (genus vectors) has no spatial geometric configuration and belongs to a topological virtual state;

under field-coupled evolution, genus vectors condense, become structured, and generate local manifold entities, forming measurable and observable geometric structures.

Topological information collapses into geometric matter structures.

(2) Real → Virtual: Geometric Entities Are Deconstructed into Genus Vectors

Formed geometric entities can be deconstructed, dissolved, and smoothed out in spatial configuration during dynamic evolution;

the degrees of freedom of entity geometry retract, retaining pure topological vector information and transforming into genus structures in a topological virtual state.

Geometric structures are elevated into topological information encoding.

IV. Paradigm Comparison between the Old and New Topological Systems

4.1 Classical Topological Paradigm

· Topology is fixed and genus is invariant

· Entities are always entities, and holes are always holes

· No evolution equations and no dynamics

· Applicable only to static, already-formed manifolds

4.2 DVG Dynamic Topological Ball Paradigm

· Topology can evolve, be born or annihilated, and undergo phase transitions
· Entities ↔ topological holes undergo bidirectional reversible transformation
· Uniformly governed by vector field coupling dynamics
· Encompasses static topology and extends to dynamic topology

Core breakthrough: topology is upgraded from a “geometric property” to an “independent dynamic field.”

V. Value of Theoretical Innovation

1. Breaking the century-old premise of topological invariance

It overturns the inherent framework in which topology can only be analyzed statically and establishes a dynamic topological phase-transition system.

2. Unifying the dynamic sources of geometry and topology

Gravitational geometric evolution and topological structure evolution share the same origin, the same equation, and the same mechanism.

3. Establishing a new mechanism for virtual–real topological transformation

It realizes for the first time the reversible transformation between geometric real structures and topological virtual structures, filling a gap in modern topology.

4. Constructing an extensible new branch of topology

Classical topology = static topology

DVG topology = dynamic evolutionary topology

VI. Scope of Application and Compatibility Statement

1. Full compatibility with classical topology

When topological evolution is frozen and genus vectors do not condense or deconstruct, DVG topology automatically degenerates into the classical Poincaré topological system.

2. Expanding the applicable boundary of original topology

This model is applicable to scenarios that classical topology cannot describe, such as manifold generation, topological phase transitions, birth and annihilation of spacetime structures, and dynamic hole evolution.

VII. Conclusion

This paper establishes a dynamic topological model of the topological ball within the Dynamic Vector Geometry (DVG) system, introduces genus vector dynamics, and realizes bidirectional virtual–real topological transformation. It upgrades the originally rigid, static, and invariant classical topology into a dynamic topological system that can evolve, be born or annihilated, undergo phase transitions, and be mutually transformed.

This model greatly expands the theoretical boundary of modern topology and forms a two-tier complete geometric system consisting of static topology plus dynamic topology, providing a new mathematical-physical foundation for high-dimensional topological evolution, the origin of spacetime topology, and phase transitions of geometric structures.