Heegaard Floer Homology and Dehn Surgery Constraints on Smooth 4-Manifolds: A Transfinite Approach

Mathematical Applications of Science Fiction We establish a rigorous obstruction theory for smooth4-manifolds bounded by 3-manifolds obtained via Dehnsurgery on knots in S3. By analyzing the mapping cone ofthe Heegaard Floer surgery exact triangle and introducinga transfinite filtration on the Spinc cobordism maps, wederive a generalized inequality relating the d-invariants ofthe surgery duals to the intersection form of the bounding4-manifold. Specifically, we prove that if a rational homol-ogy sphere Y obtained by p/q-surgery bounds a smooth,negative-definite 4-manifold W , the correction terms of Ymust satisfy a lattice-theoretic embedding condition gov-erned by the Donaldson diagonalization theorem, refined bythe Ozsváth-Szabó grading shifts. We further extend thisto a hyper-filtration category, providing constraints on theexistence of symplectic fillings.