Canonical Neighborhoods for 4-Dimensional Ricci Flow with Surgery and the Topology of Manifolds with Positive Isotropic Curvature

This monograph presents a rigorous analysis of the Ricci flow with surgery on compact four-dimensional Riemannian manifolds satisfying the positive isotropic curvature (PIC) condition. We systematically construct the surgery algorithm by establishing the existence of canonical neighborhoods—specifically -necks, -caps, and components of positive sectional curvature—in the high-curvature regions of the flow. By extending the Hamilton-Perelman estimates to dimension four, we prove that the PIC condition is preserved under surgery and that the flow becomes extinct in finite time. Consequently, we derive the topological classification of such manifolds, demonstrating that any compact 4-manifold with positive isotropic curvature and no essential incompressible space-form is diffeomorphic to a spherical space form, S3 S1, or a connected sum thereof. The analysis bridges the gap between geometric evolution equations and the smooth topology of 4-manifolds, refining the classification results of Hamilton, Chen, and Zhu.