# Topological Differences between Riemann Geometry and the Alpha Group via Graph Methods

Cleber Souza Corrêa, Thiago Braido Nogueira de Melo

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Cleber Souza Corrêa, Thiago Braido Nogueira de Melo. Topological Differences between Riemann Geometry and the Alpha Group via Graph Methods. 2025. hal-05281857

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# Topological Differences between Riemann Geometry and the Alpha Group via Graph Methods

Cleber Souza Corrêa

Institute of Aeronautics and Space, São José dos Campos, SP, Brazil

E-mail: clebercsc@fab.mil.br

Thiago Braido Nogueira de Melo

Institute of Aeronautics and Space, São José dos Campos, SP, Brazil

E-mail: braidotbnm@fab.mil.br

September 23, 2025

## Abstract

This work presents a comparative topological analysis between Riemannian geometry and the Alpha group structure using graph-theoretic methods. The antisymmetry of the matrix $M_{\theta}$ plays a central role in generating internal vectorial dynamics, giving rise to persistent and global topological paths that are absent in classical Riemannian manifolds. Through homological analysis of graphs induced by $M_{\theta}$, we demonstrate that the Alpha group exhibits a robust and dimension-independent topology, with fundamental invariants $H_0$ and $H_1$ that never collapse, even for high-dimensional nontrivial spheres. Moreover, universal linear relations between homologies hold regardless of the specific block of $M_{\theta}$ under consideration, indicating that the underlying connectivity of space is preserved across all configurations. These results establish the Alpha group framework as a coherent and globally consistent alternative to the metric-limited structures of Riemannian geometry.

**MSC 2020:** Primary 53C05, 55N35; Secondary 05C90, 81T13.

**Keywords:** Alpha group geometry, Riemann geometry, non-Riemannian geometry, graph theory, homology persistence, geometric dynamics.

## 1 Introduction

In his book Space, Time, Matter [26], Hermann Weyl explored the foundations of the metric continuum by extending the geometry of Riemann. In Chapter 2, he described how the metric itself defines distances and paths. However, this geometry remains inherently limited, since it constrains exploration to local neighborhoods, restricting the dynamics to bounded regions of the manifold. Weyl’s visionary contribution was to recognize that such a structure could be generalized through the notion of gauge, in which local symmetries naturally evolve into global ones. This historical insight resonates with our present construction, while the Riemannian framework embodies the restrictions of local dynamics. The articles [6] [7] [9] [10] describe the Alpha group as a framework that enables global connectivity and exploration, overcoming the limitations of purely metric approaches. They also represent a dynamic internal geometry
framework with complex eigenvalues and a dominant imaginary part. In the paper on the Alpha
group metric [7], its algebraic structure was defined in $\mathbb{R}^4$ space, proposing a new interpretation
of the geometry and topology of hypercomplex space. This group was described by a tensor
metric formula that generalizes the classical Riemannian and Euclidean metrics. From this
formulation, it was demonstrated that the Riemannian and Euclidean metrics can be seen as
particular cases of the Alpha group metric. This suggests that the Alpha group offers a more
general and unified framework for describing distances and curvatures in higher-dimensional
spaces.

The article [9] investigates the dynamical behavior of a system of ordinary differential equa-
tions (ODEs) governed by a matrix representing the division operation in the Alpha group alge-
bra. Through numerical simulations using the fourth-order Runge-Kutta method implemented
in Python, the authors observe topological transitions induced by the matrix, controlled by a
rotational parameter. The results reveal the emergence of topological knots and the existence
of critical points at which the rotation between division planes transitions from 0 to $\pi/2$ ra-
dians. Near zero radians, the system exhibits a Euclidean geometric structure, while rotations
close to $\pi/2$ define an Alpha group space. In these transitions, the system of ODEs exhibits
qualitative changes in dynamical behavior, reflecting distinct topological behaviors. The Alpha
Group matrix is interpreted as a generator of symmetry transformations, potentially analogous
to gauge fields under local or global symmetries, providing a computational framework for
exploring dynamical topologies, attractors at infinity, and internal coherence in hypercomplex
vector spaces. This paper [10] develops the formulation of the 4D geometry of the Alpha group
based on the matrix $M_\theta$, constructed using De Moivre's formula and the Kronecker product.
This matrix admits a decomposition into symmetric and antisymmetric parts, describing trans-
formations such as dilation, contraction, and shearing in four dimensions. A central element is
the imaginary operator $\mu$, defined as a topological and idempotent projector ($\mu^2 = \mu$), acting as
an infinity operator that compactifies $\mathbb{R}^4$ into nontrivial $S^4$ and guarantees the global coherence
of the structure. This formalism allows us to interpret the transition between Euclidean space
($\theta \approx 0$) and a compactified hyperbolic regime ($\theta \approx \pi/2$), associating the Alpha group with
fibers, bundles, and gauge principles in analogy with non-abelian theories.

The methodology combines algebraic analysis and Monte Carlo simulations, demonstrating
that variations of $\theta$ lead to topological transitions and dynamic changes, including attractors,
Hopf bifurcations, and coherent/chaotic regimes. The results highlight that the geometry of
the Alpha group incorporates robust internal symmetries, transitions between local and global
regimes, and compactification properties in the asymptotic limit.

In contrast to classical Riemannian geometry, which relies on a metric tensor defining dis-
tances and curvature locally, the Alpha group framework suggests the emergence of a non-
Riemannian structure. The presence of the idempotent operator $\mu$, acting as a topological
projector. Introduces anisotropies and compactification effects that standard Riemannian man-
ifolds cannot capture. Instead of depending solely on local curvature, the geometry acquires
global coherence through projective closure and algebraic constraints, placing it closer to non-
Riemannian approaches such as Weyl geometry [23] [27] and certain models in gauge the-
ory [24] [25] [29]. This perspective highlights that the Alpha Group algebra extends beyond
metric-based formulations, offering a richer framework where symmetry, antisymmetry, and
topological invariants govern the dynamics of space. Riemann geometry, while powerful in
describing local metrics and manifolds, has limitations in capturing global topological dynam-
ics. The Alpha group, through its matrix $M_\theta$ and intrinsic antisymmetry, reveals higher-order
structures that enable exploration of persistent paths and complex topology. This work aims
to map these differences using graph-based representations. The contributions of Donaldson
and Kronheimer in the theory of four-manifolds [13] [12] and Connes' formulation of non-
commutative geometry [4] provide two complementary perspectives that illuminate the role of
coherence as a fundamental structural principle. In the geometric setting, the study of bundles
and anti-self-dual connections in four dimensions revealed that smooth structures are not ar-
bitrary but constrained by internal consistency, establishing intrinsic regimes of coherence on
nontrivial $S^4$. In the algebraic setting, noncommutative geometry demonstrated that spaces may
be encoded by operator algebras, in which idempotents act as structural projectors that replace
classical points with coherent modules. Both frameworks converge on the idea that absolute
freedom is replaced by intrinsic mechanisms of selection. This perspective resonates with the
Frobenius approach [14] [19], where idempotent conditions govern the emergence of coherent
structures. This paper presents a solid case, considering a framework with distinctive topo-
logical properties that surpass those of classical Riemannian geometry in terms of global and
dynamic exploration of space. In this regard, it aims to demonstrate the fundamental difference
between the two frameworks: the transition from a restricted surface geometry (Riemannian)
to a dynamical space geometry (Alpha group). Alpha group geometry is not defined by a fixed
surface (sphere) but by algebraic operations (the matrix $M_{\theta}$) that generate dynamics. The ma-
trix $M_{\theta}$ is not static; it varies with the parameter $\theta$, and its antisymmetry generates a kind
of vector field or internal flux within the space. The complex eigenvalues (with a dominant
imaginary part) are the mathematical signature of this oscillatory and nontrivial dynamics. It
must be emphasized that the actual object of study is not the classical sphere $S^n$, but rather the
directed graph induced by the matrix $M_{\theta}$. The matrix $M_{\theta}$, which contains the idempotent oper-
ator $\mu$, is employed to generate a combinatorial structure that can be represented as a directed
graph. In this construction, the vertices correspond to algebraic states associated with the blocks
of $M_{\theta}$, while the oriented edges encode antisymmetric relations and internal dynamical flows
within the Alpha group algebra. Thus, the homological analysis is carried out on this induced
graph, reflecting the intrinsic algebraic and topological dynamics of the nontrivial system. In
this work, the expression "nontrivial sphere" refers not to the classical smooth manifold $S^n$
studied in algebraic topology but to the discrete topological structure induced by the operator
$M_{\theta}$. Each block of $M_{\theta}$ generates an adjacency graph, whose vertices and oriented edges encode
algebraic states and antisymmetric relations. The resulting Betti numbers therefore arise from
this graph-induced topology, rather than from the intrinsic homology of the continuous sphere.
The terminology "nontrivial sphere" is employed to emphasize that the induced structure re-
tains persistent paths and cavities absent in the Riemannian case, while being fundamentally
combinatorial in nature. Consequently, any comparison with the classical homology of $S^n$ must
be understood as an analogy, not as a strict equivalence.

# 2 Methodology

## 2.1 Graph-Based Methodology

The methodology employed in this work relies on the representation of topological and alge-
braic structures through graph theory [1] [11] [28]. In this approach, each vertex of the graph
encodes a fundamental state or element of the matrix $M_{\theta}$. At the same time, the directed edges
capture the antisymmetric relations that arise from the system's internal dynamics. By con-
structing such directed graphs, one can visualize and analyze how local interactions between
elements are organized, providing a discrete and combinatorial interpretation of continuous
structures that are otherwise difficult to capture analytically. This graphical framework allows
us to identify cycles, connectivity patterns, and the persistence of homological features across
scales.

Beyond visualization, the graph-based methodology serves as a computational tool to explore the dynamics of $M_\theta$. By associating weights and orientations to the edges, it becomes possible to model the internal geometric dynamics as flows on the graph, in which the imaginary component plays a dominant role. Homological analysis of these graphs reveals invariant features such as Betti numbers, which encode universal properties of connectivity and higher-dimensional cycles. In this way, the graph structure not only represents the antisymmetric dynamics of the Alpha group but also provides a bridge between the local restrictions of Riemannian geometry and the global exploratory capacity of the generalized topological framework.

## 2.2 Graph Construction

* **Riemann:** nodes represent points on the manifold (e.g., a sphere), and edges connect nearest neighbors, simulating local geodesics.

* **Alpha group:** nodes correspond to topological positions derived from $M_\theta$; edges reflect oriented connections and signs derived from the antisymmetry, representing internal vectorial dynamics.

The matrix $M_\theta$ [6] [7] [8] [10], with its terms on the main diagonal $\{1, i, \mu, i.\mu\}$, concretely represents the algebra of the Alpha group. The diagonal elements anchor the essence of each mode of the system with its intrinsic properties (including imaginary aspects and the topological invariant $\mu$). This represents two simultaneous dynamics: one real (rotation) and the other imaginary (distortion or bifurcation). The matrix $M_\theta = A(\theta)$. $B(\mu)$ appears to be a non-Hermitian structure, with explicit dependence on the angular parameter $\theta$, and can be seen to generate internal vector variations. In the formulation of the Alpha group algebra, which can be defined as the multiplication of an angular matrix $A(\theta)$ by another phase/amplitude matrix $B(\mu)$.

$$ M_{\theta} = \begin{pmatrix} 1 & -\cot\theta & -\tan\theta & 1 \\ \cot\theta & 1 & -1 & -\tan\theta \\ \tan\theta & -1 & 1 & -\cot\theta \\ 1 & \tan\theta & \cot\theta & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & i & 1 & 1 \\ 1 & 1 & \mu & 1 \\ 1 & 1 & 1 & i.\mu \end{pmatrix} \quad (1) $$

The Alpha group matrix $M_\theta$ can be interpreted as a structure that exhibits a coherent dynamic associated with the fibers of its complex eigenvalues. At certain critical points, especially for:

$$ \theta_n = n \cdot \pi / 2, \quad n \in \mathbb{Z} $$

Regular discontinuities (smooth bifurcations or transitions), qualitative changes in the topological structure, and the emergence of dominant complex oscillatory solutions occur, coherently organizing space. These points $\theta_n$ function as natural rhythms of coherent alignment of fibers in higher-dimensional space.

## 2.3 Methodology for Betti Number Estimation via Graphs

In this work, we adopt a purely discrete, graph-based approach to estimate the Betti numbers $H_0, H_1$, and $H_2$ of spheres $S^n$ associated with the operator $M_\theta$.
### 2.3.1 Graph Construction

Given an operator $M_\theta \in \mathbb{C}^{(n+1) \times (n+1)}$, we construct an adjacency graph $G = (V, E)$ by applying a threshold $\epsilon > 0$:

$$A_{ij} = \begin{cases} 1 & \text{if } |(M_\theta)_{ij}| > \epsilon \text{ and } i \neq j, \\ 0 & \text{otherwise.} \end{cases}$$

Vertices correspond to the indices of $M_\theta$, and edges are defined by the nonzero entries of the adjacency matrix $A$.

## 2.4 Betti Number Estimation

**$H_0$ - Connected Components:** The number of connected components of $G$ directly gives $H_0$.

**$H_1$ - Independent Cycles:** Using Euler's characteristic for graphs, the number of independent 1-dimensional cycles is estimated as:

$$H_1 = |E| - |V| + H_0,$$

where $|V|$ and $|E|$ are the number of vertices and edges in $G$.

**$H_2$ - 2D Cavities:** 2-dimensional cavities are approximated by counting triangles (3 clicks) in the graph.

## 2.5 Remarks

This methodology allows the estimation of Betti numbers from the operator $M_\theta$ without relying on specialized homology software. By remaining purely graph-based, it ensures computational simplicity, generality, and reproducibility for nontrivial spheres of arbitrary dimension. Each block in the Alpha group construction is understood as a composition of matrices $M_\theta$ that share specific internal properties, such as symmetries or asymmetry relations. Rather than being a simple submatrix, each block represents a larger dynamical unit, formed by the set of these matrices, whose interactions define the global topology of the system. The aggregation of multiple blocks allows the formation of complex structures in the dynamical graph, directly reflecting the cycles and cavities observable in the persistent homology of the Alpha group. Comparative tests were performed between the pure graph method and path homology. The results demonstrate the robustness of the pure graph approach in the topological analysis of the matrices $M_\theta$. Comparing the Betti numbers obtained via path homology [2] [3] [20] [18], we observe complete agreement in the values of $H_0$, $H_1$, and $H_2$, indicating that the topological invariants are preserved regardless of the adaptation to account for directed cycles. This alignment shows that a methodology based exclusively on graphs is sufficient to capture the essential homological structure of the system, without resorting to the more complex path homology techniques. Consequently, the pure graph method proves to be not only simple but also reliable and consistent for the topological analysis of discrete dynamics, ensuring stable and invariant results across variations in system parameters.
# 3 Results

## 3.1 Nontrivial homological mapping induced by $M_\theta$.

The $4 \times 4$ matrix $M_\theta$ exhibits antisymmetry and eigenvalue fibers that generate internal topological dynamics. Its structure ensures the persistence of $H_0$ and $H_1$ and directs global topological paths. The decomposition of the matrix $M_\theta$ into blocks provides a natural way to extend the homological analysis to different dimensions. Each block acts as a local connectivity generator, whose adjacency patterns, when interpreted as a graph, directly induce a set of Betti numbers. Thus, the homological mapping emerges from the matrix structure itself: the blocks of $M_\theta$ serve as elementary units which, when coupled, produce new connectivity and resonance relations. The homology groups computed in this context do not correspond to those of the classical $n$-dimensional sphere $S^n$. Instead, they arise from the graph structure induced by the matrix $M_\theta$, which encodes the topological and geometric relationships between the system's components. As a result, the observed homology features reflect the connectivity and higher-order interactions present in the discrete graph, rather than the continuous manifold of a nontrivial sphere. This distinction is crucial, as it implies that any direct comparison with the nontrivial homology of $S^n$ may be misleading. Therefore, the interpretation of these homology groups should be made in the context of the induced graph topology, recognizing that phenomena such as cycles, cavities, and higher-dimensional holes correspond specifically to the combinatorial structure defined by $M_\theta$.

In this way, the topology is not externally imposed but revealed by the internal dynamics of the operator, making $M_\theta$ a universal generator of homological patterns. It is important to stress that the homology associated with $M_\theta$ should not be confused with the classical homology of the spheres $S^n$. While the latter arises from the continuous manifold structure of nontrivial $S^n$, the former emerges from a combinatorial and graph-induced topology. In this framework, homological invariants are defined not by the intrinsic geometry of the sphere but by the adjacency relations and antisymmetric connections generated by the matrix $M_\theta$. As a result, cycles, connected components, and higher-dimensional cavities must be interpreted as graph-theoretic features intrinsic to the algebraic dynamics of the Alpha group, rather than as classic topological invariants of $S^n$ itself.
Directed graph highlighting antisymmetry of ($M_θ$)

<figure><img src="image_2.png" /><figcaption>Figure 1: Alpha Group graph: global paths, orientation, and antisymmetry evident.</figcaption></figure>

Figure 1 shows an antisymmetric directed graph structure that represents and models the structure of the matrix $M_θ$. It presents an antisymmetric dynamics of the matrix that has an internal geometric dynamics vector with complex values, where the dominant part is the imaginary part.

### 3.1.1 Homology Groups and Dominant Coherence of the Alpha group

The analysis of the homology groups [4], [17], [15] of the Alpha Group, performed using a graph-theoretic method [11] [17] [21], reveals a highly organized internal structure, manifested in the Betti numbers ($H_0, H_1, H_2$) associated with each sphere $S^n$ [16] [17] [22]. In this framework, each node of the graph represents a sphere, while edges reflect the topological interactions and the decomposition of the group into blocks. It is worth noting that the linear relation $H_2 = 4H_0$, although obtained empirically from numerical simulations, can be theoretically justified by the block structure of $M_θ$. Each additional block contributes one connected component ($H_0 \mapsto H_0 + 1$) and simultaneously introduces; Its antisymmetric connections, generating four independent triangular cavities. This fixed ratio arises from the combinatorial symmetry of the 4 × 4 block structure, in which every component counted in $H_0$ necessarily anchors four two-dimensional cycles. Hence, the relation $H_2 = 4H_0$ is not accidental but a structural property of the induced graph topology.

For lower dimensions ($S^2$–$S^7$), $H_1$ and $H_2$ exhibit irregular variations, indicating only partial dynamic coherence. However, starting from $S^{15}$, the Betti numbers follow fixed linear proportions ($H_1 = 3H_0$, $H_2 = 4H_0$), and the block size remains constant, demonstrating *dominant coherence* and a universal pattern in the group’s action. This behavior is sustained by the *topo-
logical anchor $\mu \in S^4$, an idempotent projective operator, which ensures that each substructure integrates smoothly and that the group dynamics maintain global consistency. Consequently, the Alpha group's structure in higher dimensions can be represented linearly and predictably.

<table><caption>Table 1: Homology groups and block structure of the Alpha Group obtained via the graph method: number of blocks of $M_\theta$ associated with different nontrivial graph-induced topologies $S^n$.</caption><thead><tr><th>$S^n$</th><th>$n$</th><th>$H_0$</th><th>$H_1$</th><th>$H_2$</th><th>num_blocks</th><th>block_size</th></tr></thead><tbody><tr><td>$S^2$</td><td>2</td><td>1</td><td>1</td><td>1</td><td>1</td><td>3</td></tr><tr><td>$S^3$</td><td>3</td><td>1</td><td>3</td><td>4</td><td>1</td><td>4</td></tr><tr><td>$S^4$</td><td>4</td><td>2</td><td>1</td><td>1</td><td>2</td><td>3</td></tr><tr><td>$S^5$</td><td>5</td><td>2</td><td>2</td><td>2</td><td>2</td><td>3</td></tr><tr><td>$S^6$</td><td>6</td><td>2</td><td>4</td><td>5</td><td>2</td><td>4</td></tr><tr><td>$S^7$</td><td>7</td><td>2</td><td>6</td><td>8</td><td>2</td><td>4</td></tr><tr><td>$S^{15}$</td><td>15</td><td>4</td><td>12</td><td>16</td><td>4</td><td>4</td></tr><tr><td>$S^{23}$</td><td>23</td><td>6</td><td>18</td><td>24</td><td>6</td><td>4</td></tr><tr><td>$S^{31}$</td><td>31</td><td>8</td><td>24</td><td>32</td><td>8</td><td>4</td></tr></tbody></table>

As illustrated in Figure 2, the Betti numbers $H_1$ and $H_2$ evolve irregularly for $S^3-S^7$, reflecting partial coherence, and then increase linearly from $S^{15}$ onward, signaling the emergence of dominant coherence as the number of blocks grows.

<figure><img src="image_3.png" /><figcaption>Figure 2: Progression of Betti numbers $H_1$ and $H_2$ versus the number of blocks in the Alpha Group. The yellow region indicates partial coherence ($S^3 - S^7$), while the green region highlights dominant coherence ($S^{15} - S^{31}$).</figcaption></figure>

It is important to emphasize that the homology groups $(H_0, H_1, H_2)$ presented here do not correspond to the intrinsic classical homology of the spheres $S^n$, but rather to the homology of the block structure induced by the matrix $M_\theta$. Specifically, $H_0$ counts the connected components of the graph, $H_1$ identifies independent cycles formed by block connections, and $H_2$ captures two-dimensional cavities arising from triangular arrangements within the graph.
<figure><img src="image_4.png" /><figcaption>Figure 3: Normalized progression of Betti numbers $H_0$, $H_1$, and $H_2$ versus the number of blocks. Partial coherence is observed for $S^3 - S^7$, while $H_0$ demonstrates universal growth across all systems.</figcaption></figure>

In Figure 3, $H_0$ (connected components) shows complete coherence; all systems ($S^3-S^7$) behave identically, with linear and predictable growth. Even at $H_0$, the Alpha group surpasses Riemannian geometry, revealing a deeper universal connectivity encoded in its homological structure. This indicates a universal mechanism for adding new components. $H_1$ (cycles) exhibits high coherence despite fluctuations; all systems follow the same general trend, suggesting that a universal process also governs tunnel formation.

In contrast, $H_2$ voids/cavities display irregular behavior for lower dimensions: $S^3$ produces almost no voids, $S^5$ and $S^6$ create moderate amounts, while $S^7$ generates the most voids earliest. This demonstrates that higher-order topological complexity emerges as a system-specific property. From $S^{15}$ onward, however, dominant coherence reappears, reinforcing the universality of the Alpha group's homology.

Overall, this analysis reveals that the Alpha Group exhibits a smooth, manifold-like structure in higher dimensions, with homological features governed by the projective operator $\mu$ and a universal block-growth mechanism.

<table><caption>Table 2: Homology results of the Alpha group for different conditions, dimensions, and parameters. $H_0$ and $H_1$ remain invariant at critical points.</caption><thead><tr><th>Condition</th><th>S<sup>n</sup></th><th>θ</th><th>μ</th><th>H<sub>0</sub></th><th>H<sub>1</sub></th><th>H<sub>2</sub></th><th>Regime</th></tr></thead><tbody><tr><td>Critical (π/2)</td><td>S<sup>15</sup></td><td>1.5708</td><td>1.0</td><td>4</td><td>4</td><td>0</td><td>Chaotic</td></tr><tr><td>Critical (π/2)</td><td>S<sup>3</sup></td><td>1.5708</td><td>1.0</td><td>1</td><td>1</td><td>0</td><td>Chaotic</td></tr><tr><td>Critical (π/2)</td><td>S<sup>7</sup></td><td>1.5708</td><td>1.0</td><td>2</td><td>2</td><td>0</td><td>Chaotic</td></tr><tr><td>Pre-critical</td><td>S<sup>15</sup></td><td>1.4000</td><td>1.0</td><td>4</td><td>12</td><td>16</td><td>Coherent</td></tr><tr><td>Pre-critical</td><td>S<sup>3</sup></td><td>1.4000</td><td>1.0</td><td>1</td><td>3</td><td>4</td><td>Coherent</td></tr><tr><td>Pre-critical</td><td>S<sup>7</sup></td><td>1.4000</td><td>1.0</td><td>2</td><td>6</td><td>8</td><td>Coherent</td></tr><tr><td>Post-critical</td><td>S<sup>15</sup></td><td>1.6000</td><td>1.0</td><td>4</td><td>12</td><td>16</td><td>Coherent</td></tr><tr><td>Post-critical</td><td>S<sup>3</sup></td><td>1.6000</td><td>1.0</td><td>1</td><td>3</td><td>4</td><td>Coherent</td></tr><tr><td>Post-critical</td><td>S<sup>7</sup></td><td>1.6000</td><td>1.0</td><td>2</td><td>6</td><td>8</td><td>Coherent</td></tr><tr><td>High μ</td><td>S<sup>15</sup></td><td>1.5700</td><td>2.0</td><td>4</td><td>4</td><td>0</td><td>Chaotic</td></tr><tr><td>High μ</td><td>S<sup>3</sup></td><td>1.5700</td><td>2.0</td><td>1</td><td>1</td><td>0</td><td>Chaotic</td></tr><tr><td>High μ</td><td>S<sup>7</sup></td><td>1.5700</td><td>2.0</td><td>2</td><td>2</td><td>0</td><td>Chaotic</td></tr><tr><td>Low μ</td><td>S<sup>15</sup></td><td>1.5700</td><td>0.1</td><td>4</td><td>4</td><td>0</td><td>Chaotic</td></tr><tr><td>Low μ</td><td>S<sup>3</sup></td><td>1.5700</td><td>0.1</td><td>1</td><td>1</td><td>0</td><td>Chaotic</td></tr><tr><td>Low μ</td><td>S<sup>7</sup></td><td>1.5700</td><td>0.1</td><td>2</td><td>2</td><td>0</td><td>Chaotic</td></tr></tbody></table>
Table 2 presents a significant topological result, where $H_0$ represents the connected components, and its persistence indicates that the structure of the space never completely fragments, even in a chaotic regime. However, $H_1$ represents first-dimensional cycles (loops or holes), and the fact that it does not collapse shows that fundamental topological paths remain preserved, even when local coherence is minimal. The implications of maintaining $H_0$ and $H_1$ at critical points indicate that there are always structural paths and connections available in topological space, allowing, for example, the preservation of invisible loops or bundles. The results presented in this study primarily consider the case of 4 blocks of the matrix $M_\theta$. However, the analysis suggests that, as the number of blocks increases, $H_2$ does not stabilize in absolute value but rather follows a linear growth, obeying $H_2 = 4H_0$. Thus, the expression 'tends to stabilize' should be understood as the stabilization of this linear pattern (i.e., the constant ratio $H_2/H_0 = 4$). while $H_0$ and $H_1$ remain invariant. This behavior reinforces the topological universality of the Alpha group, indicating that its coherent and non-trivial structure is preserved across larger scales and in critical regimes. This reinforces the robustness of the Alpha group, which maintains topological invariants even under extreme conditions. In this context, we can observe a universal pattern; this invariance in $H_0$ and $H_1$ does not depend on the dimension $S^n$ ($S^3, S^7, S^{15}, S^{31}$), which reinforces the universality of the behavior of the Alpha group and its matrix $M_\theta$. The collapse of $H_2$ shows that second-dimensional holes are not structural and can vanish near critical regimes ($\theta \approx n\pi/2$). Here, the term *collapse of $H_2$* does not imply a total annihilation of the structure, but rather the loss of persistence of two-dimensional cavities. These cavities may appear in certain regimes (pre- and post-critical), but they vanish in critical regimes ($\theta \approx n\pi/2$). However, we further observe that, when considering the antisymmetric contribution of the matrix $M_\theta$, the homological behavior exhibits a distinct feature: topological coherence tends to be preserved, so that the collapse of $H_2$ may not occur, in contrast with the purely symmetric case. Therefore, the “collapse” should be understood as a structural instability of $H_2$, in contrast to $H_0$ and $H_1$, which remain universal invariants of the Alpha group topology.

The homology of the Alpha group exhibits remarkable universality across various dimensions. The results, obtained from multiple graphs induced by the matrix $M_\theta$, show that the fundamental invariants $H_0$ and $H_1$ persist even under critical conditions and extreme variations of $\mu$, highlighting the robust and dimension-independent nature of the topological structure. This universality suggests that essential pathways and cycles are preserved, providing a coherent framework that underlies the Alpha Group's topology and enables the reliable exploration of higher-dimensional fibrations and invisible loops.

### 3.1.2 Analysis Summary and Pattern Recognition

The analysis of the operator $M_\theta$, structured in blocks derived from the fundamental matrix $M_\theta$, using a graph-based methodology, allowed the identification of the fundamental homological signature of the *Alpha Group*. The results show that:

* The number of connected components $H_0$ never collapses, indicating that the topological structure generated by the operator always preserves a minimal connectivity among the blocks of $M_\theta$.

* The first Betti number $H_1$ also never collapses, revealing the persistent presence of cycles and recurring flows, which characterize the internal dynamics of the operator, as never being connected.
* The second Betti number $H_2$ may collapse across various blocks, showing that two-dimensional cavities are not a structural invariant of the system but rather an unstable property that emerges or disappears depending on the block composition. It is important to note that $H_2$ collapses in the symmetric part of the matrix $M(\theta)$, whereas the antisymmetric contribution preserves the topological coherence.

## 3.2 Graph-based Comparison

Figure 4 represents a 3D sphere with a transparent blue mesh. The sphere symbolizes a local space (with a local metric) and geometric constraints. Its paths between points are limited, meaning there are restrictions on exploring the space. This suggests that the system's dynamics are more local and restricted, limited by the metric properties of Riemannian geometry. In the case of Figure 4 on the right, with the Alpha group representation, the graph represents the higher structure of the Alpha group, in which there is more freedom of movement and interconnection between points. It does not have the restriction of local space; the colors of the lines can indicate different types of connections or internal dynamics of the system. Unlike the sphere on the left, it is possible to explore the space globally, not just locally. This illustrates a more abstract, dynamic topological structure that enables connections and global exploration. Visually, the difference is clear; the sphere closes paths, while the Alpha Group graph connects points and allows for greater exploration. It is essential to note that the comparison with classical homology of spheres can be misleading. In our framework, the Betti numbers are not computed from the smooth manifold $S^n$, but rather from graphs induced by the operator $M_\theta$. In this sense, the term *nontrivial sphere* is adopted to stress that the resulting topological structure does not coincide with the standard homology of $S^n$, but instead emerges as a discrete, graph-based analogue. Thus, any apparent similarity with classical spheres should be interpreted as a graph-induced structural correspondence rather than as a strict topological equivalence.

<figure><img src="image_5.png" /><figcaption>Figure 4: shows the difference between Riemann geometry and Alpha group geometry</figcaption></figure>

A strictly topological analysis reveals the structural limitations of Riemannian geometry; it exhibits local boundedness, its geodesics are confined to the manifold of the sphere, and its homological invariants show limited persistence, with $H_0$ fixed and $H_1$ collapsing for $n > 1$

In the case of the Alpha group, a superior structure emerges. The matrix $M_\theta$ exhibits antisymmetry, generating an internal vectorial geometric dynamic with eigenvalue fibers whose imaginary parts dominate. This allows for global and persistent topological paths in $\mathbb{R}^4$ with invariant $H_0$ and $H_1$.
<table><caption>Table 3: Comparison between the classical homology of the sphere $S^n$ and the nontrivial homology induced by the block structure of $M_\theta$.</caption><thead><tr><td>Aspect</td><td>Classical Homology of S<sup>n</sup></td><td>Nontrivial Homology Induced by M<sub>θ</sub></td></tr></thead><tbody><tr><td>Reference space</td><td>Continuous manifold (surface of the sphere).</td><td>Directed graph from M<sub>θ</sub> (blocks and adjacencies).</td></tr><tr><td>Topological generators</td><td>Curves and cycles defined by a Riemannian metric.</td><td>Oriented connections and cycles induced by matrix antisymmetry.</td></tr><tr><td>Homology groups</td><td>H<sub>0</sub> = 1, H<sub>1</sub> = 0 for n > 1, H<sub>n</sub> = Z (others vanish).</td><td>H<sub>0</sub> and H<sub>1</sub> always persist; H<sub>2</sub> may collapse or emerge.</td></tr><tr><td>Persistence</td><td>Fixed, determined by dimension n.</td><td>Varies with θ, μ, and block number; reveals transitions.</td></tr><tr><td>Universality</td><td>Limited (each S<sup>n</sup> has its own homology).</td><td>Linear relations: H<sub>1</sub> = 3H<sub>0</sub>, H<sub>2</sub> = 4H<sub>0</sub> in higher dimensions.</td></tr><tr><td>Interpretation</td><td>Static, purely metric structure.</td><td>Dynamic structure, guided by antisymmetry and projective operator μ.</td></tr></tbody></table>

Classical persistent homology analyzes the stability of holes and voids in static surfaces, examining how topological features survive changes in scale. In the Alpha group, this paradigm is inverted; the focus is not on a fixed shape but on a transformation matrix $M_\theta$ that continuously generates a dynamic connectivity graph. The observed persistence reflects not the geometry of a surface but the topology inherent to the system's own dynamics.

Even under variations of $\theta$ that drive the system into chaotic states, the fundamental invariants $H_0$ and $H_1$ remain intact, demonstrating that basic connectivity and essential cycles are emergent properties of the dynamics. Thus, the topological "surface" of the Alpha group is not fixed; it emerges from the dynamics itself, revealing a living network of forms and cycles whose persistence transcends classical geometry and encodes the structural freedom of the system.

# 4 Conclusion

Graph-based methods clearly reveal the topological differences between Riemannian geometry and the Alpha group. The antisymmetry of $M_\theta$ plays a central role in generating internal vectorial paths, enabling global and persistent exploration that cannot be captured within Riemannian geometry. This study provides a concise and technical foundation for understanding the topological dynamics of the Alpha group.

The homology of the Alpha group displays remarkable universality across different dimensions. Regardless of the nontrivial sphere $S^n$ considered, the fundamental invariants $H_0$ and $H_1$ persist even under critical conditions and extreme variations of $\mu$, highlighting the robust and dimension-independent character of the topological structure. Note that the constant feature is the ratio $H_2/H_0 = 4$, not the absolute value of $H_2$, which grows linearly with the number of blocks. This behavior reinforces the topological universality of the Alpha group, indicating that its coherent and non-trivial structure is preserved across larger scales and in critical regimes. However, we further observe that, when considering the antisymmetric contribution of the matrix $M_\theta$, the homological behavior exhibits a distinct feature: topological coherence tends to be preserved, so that the collapse of $H_2$ may not occur, in contrast with the purely symmetric case. This persistence suggests that antisymmetry plays a fundamental role in maintaining the structure, indicating a promising direction for future investigations. This universality demonstrates
that essential pathways and cycles are preserved, providing a coherent framework that supports
the topology of the Alpha group and enables the reliable exploration of higher-dimensional
fibrations and invisible loops.

The division operation defines a geometric dynamic with symmetric and antisymmetric functions, allowing the emergence of the dual topological value of $\mu$ in a projective structure. This duality leads to the formation of a global attractor, acting as a geometric fixed point or organizing center within the extended space of the Alpha group algebra.

Thus, the group structure encapsulates a projective and conformal dynamic that combines
algebraic operations with topological transformations, producing a coherent model of compact-
ified multidimensional geometry.

The results obtained through homological analysis of graphs induced by $M_\theta$ demonstrate
a topological universality that transcends specific dimensions and parameters. Invariants $H_0$
and $H_1$ never collapse, even for high-dimensional induced topologies, indicating that the fun-
damental connectivity of the space generated by the Alpha group is robust under variations of
$\theta$.

Universal relations are also observed; linear connections between homologies hold regard-
less of the specific block of $M_\theta$ analyzed, confirming that the induced topology is globally
consistent. The idempotent operator $\mu$ within $M_\theta$ guarantees dominant coherence, linking re-
gions of the space in a structured manner. The space is not merely a collection of vertices and
edges; it hosts internal flows that reflect the topological dynamics of the Alpha group. More-
over, the persistence of universal homological patterns suggests that the topology of the Alpha
group is intrinsic and not dependent on particular parameterizations or dimensions. This stands
in direct contrast to classical Riemannian geometry, where invariants may collapse or vanish
locally.

The analysis reveals that the $M_\theta$ blocks represent the discrete nontrivial sphere. The Betti
numbers remain consistent whether the blocks are considered separately or interconnected,
highlighting the topological universality of the structure.

The Alpha group maintains a nontrivial residual structure (induced graph). This means that
there is no topological emptiness; minimal coherence is always preserved through the operator
$\mu$. The results demonstrate that the Alpha group is more general than the Riemann geometry of
$S^n$. The classical sphere thus appears only as a particular case (or continuous boundary), while
the Alpha group describes a broader topological framework based on $M_\theta$ and $\mu$.

Finally, the Alpha group preserves far richer internal geometric dynamics. This establishes
the Alpha group framework as a foundational alternative to Riemannian geometry, opening new
perspectives for future applications in topology, geometry, and physics. The analysis presented
here has focused on independent blocks $M_\theta$. A natural extension, to be explored in subsequent
works, is the inter-block regime, in which global coherence is reinforced by connections be-
tween the blocks, leading to even more general results. The Alpha group reveals two distinct
regimes: in the symmetric regime, we observe rigid classical geometry; in the antisymmetric
regime, fluidity emerges, along with nonlinearity that preserves topological cavities. The coex-
istence of these two poles, chaos and coherence. This duality transforms the outcome, revealing
that reality sustains itself through oscillations between collapse (topological contraction) and
permanence (persistence of invariants). Such oscillatory behavior manifests as a dynamic topol-
ogy, where homological structures alternate between destruction and reconstruction, preserving
coherence across regimes. The rigorous algebraic formalization of the observed universal re-
lations will be presented in subsequent works; the present work establishes the computational
and structural foundations.
# Code Availability

All source code and scripts utilized in this study are publicly accessible at the following GitHub repository [5]: https://github.com/CleberCorrea15.

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