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.

For a finite rank projective bundle over a compact manifold, so associated to a torsion, Dixmier-Douady, 3-class, $w$, on the manifold, we define the ring of differential operators ‘acting on sections of the projective bundle’ in a formal sense. In particular, any oriented even-dimensional manifold carries a projective spin Dirac operator in this sense. More generally the corresponding space of pseudodifferential operators is defined, with supports sufficiently close to the diagonal, i.e., the identity relation. For such elliptic operators we define the numerical index in an essentially analytic way, as the trace of the commutator of the operator and a parametrix and show that this is homotopy invariant. Using the heat kernel method for the twisted, projective spin Dirac operator, we show that this index is given by the usual formula, now in terms of the twisted Chern character of the symbol, which in this case defines an element of K-theory twisted by $w$; hence the index is a rational number but in general it is not an integer.

We discuss the notion of the Kodaira dimension for symplectic manifolds in dimension 4. In particular, we propose and partially verify Betti number bounds for symplectic 4-manifolds with Kodaira dimension zero.