Ricci flow with surgery on four-manifolds with positive isotropic curvature

In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. We establish a long-time existence result of the Ricci flow with surgery on four-dimensional manifolds. As a consequence, we obtain a complete proof to the main theorem of Hamilton. During the proof we have actually provided, up to slight modifications, all necessary details for the part from Section 1 to Section 5 of Perelman’s second paper on the Ricci flow to approach the Poincaré conjecture.