Fourier Rank Conjecture
Title: Fourier Interference and the Rank Conjecture: A Wave-Based Constraint on Finite Group Actions on Spheres
Author: Orion Franklin, Syme Research Collective
Date: March, 2025
Abstract:
Abstract: The Rank Conjecture states that the rank of a finite group acting freely on an n-dimensional sphere is bounded by the cohomological dimension of the group. Traditional approaches rely on algebraic topology and homology, but in this paper, we introduce a novel Fourier-based approach. We analyze group actions through sinusoidal transformations and study their interference patterns using Fourier analysis. Our findings suggest that the growth of frequency interference imposes a natural bound on rank, providing a new heuristic constraint on the conjecture.
Introduction: Finite groups acting freely on spheres have long been studied in topology, with the Rank Conjecture providing a crucial upper bound on the rank of such groups. Classical techniques leverage homological and cohomological methods to establish these constraints. However, by modeling group actions as sinusoidal transformations, we explore whether frequency-based interference can naturally restrict free group actions and confirm known cases of the Rank Conjecture.
Fourier Representation of Group Actions: We model each group action as a transformation: T_g(x) = x + A sin(kx + φ) where A is the transformation amplitude, k is the frequency component, and φ represents phase shifts introduced by the group structure. Free group actions must preserve certain frequency alignment properties, and we hypothesize that excessive interference disrupts freeness.
Simulation Results:
1D and 2D Fourier Analysis:
Low-rank groups exhibit limited frequency dispersion, allowing free actions.
Increasing rank introduces significant interference, suggesting a natural upper bound.
Scaling to Higher Dimensions:
The critical frequency threshold for interference scales as f_critical(n) = 0.477 * n^(1/3).
This suggests that frequency dispersion grows non-linearly with dimension.
Comparison with Classical Rank Constraints:
Our Fourier-based rank limit closely aligns with classical cohomological constraints.
Predicted rank limits match previously proven cases of the Rank Conjecture for abelian and p-groups.
Non-Abelian Group Constraints:
Additional phase shifts in non-abelian structures amplify interference.
This suggests a stricter bound for non-abelian groups, requiring a modified scaling factor.
Fourier-Based Rank Conjecture Statement: We propose the following constraint: r ≤ dim H*(S^n; Z_2) / (0.477 * n^(1/3)) for abelian groups, with an additional correction factor c > 1 for non-abelian groups.
Conclusion and Future Work:
Our approach offers a novel Fourier-based perspective on the Rank Conjecture, providing an alternative mathematical framework for bounding rank. Future work includes:
Refining the formal proof using interference limits.
Exploring connections between Fourier constraints and homotopy theory.
Investigating empirical group action data for further validation.
This research suggests that wave-based methods may provide new insights into longstanding conjectures in algebraic topology.
References
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Discusses rank constraints in group cohomology, which this paper extends using wave-based methods.
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Covers cohomology of finite groups, relevant to classical approaches in the Rank Conjecture.
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Provides topological methods for studying finite group actions, which contrast with the wave-based approach.
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Introduces key algebraic topology concepts that are connected to Fourier-based interference.
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Applies symmetry constraints in physics, which relate to the interference effects described in this paper.
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Discusses fundamental Fourier-based constraints that parallel the interference model proposed.
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Provides a foundational perspective on group growth, relevant to how rank is constrained by interference.
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Discusses group actions in high-dimensional spaces, relevant to the conjectured wave-based rank constraint.
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The original Fourier theory foundation, underpinning the spectral interference approach.
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Establishes fundamental results in signal processing and information theory, relevant to structured interference.
