We develop monopole and instanton Floer homology groups for balanced sutured manifolds, in the spirit of [12]. Applications include a new proof of Property P for knots.

We define analytic torsion (X, E,H) 2 detH•(X, E,H) for the twisted de Rham complex, consisting of the spaces of differential forms on a compact oriented Riemannian manifold X valued in a flat vector bundle E, with a differential given by rE + H ^ · , where rE is a flat connection on E, H is an odd-degree closed differential form on X, and H•(X, E,H) denotes the cohomology of this Z2-graded complex. The definition uses pseudodifferential operators and residue traces. We show that when dimX is odd, (X, E,H) is independent of the choice of metrics on X and E and of the representative H in the cohomology class [H]. We define twisted analytic torsion in the context of generalized geometry and show that when H is a 3-form, the deformation H 7! H−dB, where B is a 2-form on X, is equivalent to deforming a usual metric g to a generalized metric (g,B). We demonstrate some basic functorial properties. When H is a top-degree form, we compute the torsion, define its simplicial counterpart, and prove an analogue of the Cheeger-M¨uller Theorem. We also study the twisted analytic torsion for T -dual circle bundles with integral 3- form fluxes.

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We show that the exterior derivative operator on a symplectic manifold has a natural decomposition into two linear differential operators, analogous to the Dolbeault operators in complex geometry. These operators map primitive forms into primitive forms and therefore lead directly to the construction of primitive cohomologies on symplectic manifolds. Using these operators, we introduce new primitive cohomologies that are analogous to the Dolbeault cohomology in the complex theory. Interestingly, the finiteness of these primitive cohomologies follows directly from an elliptic complex. We calculate the known primitive cohomologies on a nilmanifold and show that their dimensions can vary with the class of the symplectic form.

An odd Seiberg-Witten invariant imposes bounds on the signature of a closed, almost complex 4-manifold with vanishing first Chern class. This applies in particular to symplectic 4-manifolds of Kodaira dimension zero.