Structural and Topological Perspectives on Four-Dimensional Manifolds
DOI:
https://doi.org/10.63665/vjmxt515Keywords:
Four-dimensional manifolds, intersection forms, Seiberg-Witten invariants, Euler characteristic, smooth topologyAbstract
Four-dimensional manifolds occupy a uniquely complex position in modern topology. This paper examines their structural and topological properties through intersection forms, Betti numbers, Euler characteristics, and gauge-theoretic invariants. The primary objective is to systematically compare smooth and topological 4-manifolds using standard examples and established invariants. The study employs a descriptive-analytical methodology, drawing on foundational results from Freedman, Donaldson, and Witten to analyze topological invariants of key 4-manifolds including S⁴, CP², S²×S², K3, and the four-torus T⁴. The hypothesis posits that the divergence between smooth and topological categories in dimension four is uniquely determined by the arithmetic of intersection forms and gauge-theoretic constraints. Results demonstrate that Euler characteristics range from 0 (T⁴) to 24 (K3), while Seiberg-Witten invariants sharply distinguish diffeomorphism types within homeomorphism classes. The discussion affirms that no single invariant suffices for complete classification, and that the interplay between intersection forms and gauge theory remains the central framework for understanding 4-manifold topology. The conclusion highlights open problems including the smooth four-dimensional Poincaré conjecture.
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References
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