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August 12, 2025Open Access

Characterizing the 4-sphere, S^4

Joel Hass Robion Kirby

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Key Points

The status of the smooth 4-dimensional Poincaré Conjecture remains unresolved, impacting topology and geometry.
Current insights into the 4-sphere structure provide valuable context for ongoing mathematical discussions.
Analysis includes various attempts to prove or disprove the conjecture, showcasing its significance in higher dimensions.
The conjecture's unresolved nature draws attention from mathematicians seeking breakthroughs in topology.

Implication

This survey reveals the current state of the smooth 4-sphere and its relation to the Poincaré Conjecture, highlighting unresolved challenges.

Abstract

We survey the status of the smooth 4-dimensional Poincaré Conjecture, one of the central open problems in mathematics.

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Cite This Study

Hass et al. (2025) studied this question.

synapsesocial.com/papers/68af494dad7bf08b1ead4c9a https://doi.org/10.56994/jomp.001.001.003

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Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context:

1The Smooth Four-Dimensional Poincaré Conjecture under the Andrews–Curtis Conjecture: A Conditional Trisection–Handle RouteOriginal Research · 2026
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3A Geometric Solution and Interpretation of the Poincaré Conjecture: The Perspective of Visual Relief and Sweep-out ProtocolOriginal Research · 2026
4Infinite families of standard Cappell-Shaneson spheres2024
5Global Riemannian, Lie Group and Spectral Geometry of the Three-Sphere: Hopf Fibration, SU(2)×SU(2) Isometry Decomposition and Complete Hyperspherical Harmonic AnalysisOriginal Research · 2026 · 1 citations

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